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LIBRARY 

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zAcccssious  No,() 2^  7^'        Class  No. 


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THE                         * 

FIEST  BOOK 

OP 

ARITHMETIC. 

BY 

DANA   P.   COLBURN, 

PRINCIPAL  OF  THE  EHODS  ISLAND  STATE  NORMAL  SCHOOIi,  AITD 
AUTHOR  OF  "ARITHMETIC  AND  ITS  APPLICATIONS." 

PHILADELPHIA: 
}         H.   COWPERTHWAIT    &    CO. 
]              BOSTON:  SHEPARD,  CLARK  &  BROWN. 
1860. 

as 


(>irj(f 


Entered,  according  to  Act  of  Congress,  in  the  year  1856,  by 

DANA    P.  COLBURN, 

in   th«  Clerk's  Office  of  the  District   Court  of   the  United  States  for  the 

District  of  Rhode  Island. 

STEREOTYPED  BY  J.  rAGAIT. 


(ii) 


PEEFACE. 


The  Docimal  System  of  Numbers  is  one  of  the 
most  perfect  things  of  man's  invention.  It  is  so 
simple,  that  a  child  can  understand  it,  yet  so 
comprehensive  that  it  includes  all  possible  num- 
bers, represents  them  all  by  ten  simple  characters 
and  a  point,  and  bases  all  numerical  operations 
on  the  combinations  of  the  first  ten,  or  primitive 
numbers. 

These  primitive  combinations  can  easily  be 
determined.  1  can  be  added  to  each  number 
from  1  to  10 ;  so  can  each  of  the  first  ten  num- 
bers, and  on  these  will  depend  all  possible  com- 
binations in  addition.  For  since  3  +  2  =  5,  we 
have  30  +  20  =  50,  300  +  200  =  500,  &c.,  23  +  2 
==  25,  193  +  2  =  195,  &c. 

But  subtraction  is  so  closely  connected  with 
addition,  that,  as  far  as  the  primitive  numbers 
are  concerned,  a  knowledge  of  one  implies  a 
knowledge  of  the  other.  What,  for  instance,  are 
2  +  3  =  5,  2  from  5  =  3,  3  from  5  =  2,  5  =  3 
more  than  2,  &c.,  but  different  forms  of  expressing 
the  idea  that  5  is  made  up  of  2  and  3  ? 

.  __- 


PREFACE. 


It  is  certainly  more  philosophical  to  present 
these  various  forms  in  such  connexion  as  to 
show  their  mutual  dependence,  and  thus  secure 
thoroughness  from  the  outset,  than  to  present 
one  form  through  a  long  series  of  lessons,  and 
then  another  form  through  another  long  series 
of  lessons,  as  though  they  have  no  connexion  with 
each  other.  The  invariable  experience  of  teachers 
who  have  given  both  methods  a  fair  trial  is,  that 
elementary  addition  and  subtraction  can  thus  be 
taught  together,  with  greater  ease  than  either  can 
be  taught  by  itself. 

All  that  has  been  said  of  addition  and  subtmc- 
tion  applies  with  equal  force  to  multiplication  and 
division.  Hence,  the  varied  combinations  of  the 
primitive  numbers  ought  to  be  mastered  by  the 
pupil  before  he  attempts  those  of  the  derived 
numbers;  and  when  the  latter  are  introduced,  it 
should  be  in  such  a  way  as  to  show  their  depen- 
dence on  the  former. 

Another  reason  for  this  careful  elementary  in- 
struction is  found  in  the  fact  that  a  child  will  rea- 
dily understand  and  solve  a  problem  involving 
small  numbers,  when  a  similar  one,  involving 
large  numbers,  will  be  entirely  beyond  his  com- 
prehension. There  can  be  no  doubt,  then,  that 
his  attention  should  be  confined  to  small  num- 
bers, till  his  mathematical  powers  are  so  far 
developed  as  to  enable  him  to  use  large  numbers 
understandingly. 


r- 


PR  EFAGE. 


The  lessons  of  a  First  Book  of  Arithmetic 
should  be  based  on  such  principles. 

They  should  be  so  arranged  as  to  illustrate  in 
an  easy  and  familiar  manner  the  nature  and  uses 
of  numbers  and  of  numerical  operations,  to  call  into 
exercise  and  discipline  the  mental  powers,  form 
accurate  habits  of  thought  and  investigation, 
impart  a  just  self-reliance,  develop  a  power  of 
following  closely  the  most  rigid  reasoning  pro- 
cesses, and  lay  a  sure  foundation  for  future  pro- 
gress in  mathematical  studies. 

They  ought  to  be  simple  in  their  beginnings, 
gradual  in  their  developments,  interesting  in  their 
problems,  varied  in  their  exercises,  and  so  con- 
nected, that  each  shall  follow  naturally  from  those 
that  go  before,  and  prepare  the  way  for  those  that 
come  after. 

Moreover,  they  should  embody  such  a  variety 
and  extent  of  exercises  as  to  include  all  the  essen- 
tial principles  of  Arithmetic,  and  thus  prepare  the 
way  for  any  advanced  treatise,  and  even  give 
those  who  have  no  further  opportunity  for  study 
in  school^  such  discipline  as  will  enable  them  to 
meet  the  demands  of  real  life. 

The  author  has  endeavoured  to  prepare  this 
treatise  in  accordance  with  these  views.  In  its 
preparation  he  has  drawn  freely  from  the  ''  First 
Steps  in  I^umbers,"  and  the  "Decimal  System  of 
Numbers,"  both  issued  some  years  since,  —  the 
the  latter  of  which  w^as  written  by  himself,  and 


PREFACE. 


the  former  conjointly  with  Mr.  George  A.  Wal- 
ton, now  of  Lawrence,  Mass. 

Wlmtever  may  be  its  merits  or  defects,  it  is  the 
result  of  much  careful  thought  and  study,  of  con- 
siderable experience  as  a  Teacher,  and  of  an 
honest  eftbrt  to  arrange  such  a  course  of  lessons 
as  shall  aid  in  developing  the  youthful  mind,  and 
in  forming  correct  habits  of  study. 

DAN-A  P.  COLBUEN. 

Providence,  July,  1856. 


COLBURN'S   FIRST   PART, 


LESSOir  I. 


One  boy.     1  boy.  /  ^tm. 

One  girl.     1  girl.  /  a^u. 

One  ball.     1  ball.  /  /a/f. 

One.     1.  /.     I. 

1.  ITow  many  pictures  are  there  on  this  page? 

2.  How  many  boys  do  you  see  in  the  picture  ? 

3.  How  many  girls  do  you  see  in  the  picture  ? 

4.  How  many  balls  do  you  see  in  the  picture  ? 

5.  How  many  thumbs  have  you  on  your  right  hand  ? 

6.  How  many  thumbs  have  you  on  your  left  hand  ? 
_  __ 


8  colburn's  first  part. 


Note  to  the  Teacher. — The  pictures  are  not  designed  to  take 
the  place  of  exercises  -with  visible  objects,  but  rather  as  addi- 
tional illustrations  of  the  numbers  introduced.  With  young 
classes,  the  Teacher  should  give  such  easy  lessons  as  the  following, 
making  use  of  the  most  familiar  objects  as  counters. 

Even  though  the  pupils  have  used  numbers  somewhat,  such 
lessons  will  make  them  better  acquainted  with  their  nature,  and 
will  thus  ensure  a  more  rapid  advancement. 

Oral  Lessox. — Teacher,  taking  a  book,  asks:  "What  have  I 
in  my  hand?"  Ans. — *'A  book."  **  How  many  books?"  Ans. — 
'<  One  book."  "  How  many  pencils  do  I  show  you  ?"  Ans. — *'  One 
pencil."  "How  many  chairs  do  I  point  at?"  A?is.  —  "One 
chair."     "  How  many  desks  ?"     Ans. — "  One  desk." 

Tell  the  class  to  point  to  one  boy,  to  one  girl,  to  one  windoV, 
&c.,  &c. 

The  mark  1  (making  it  on  the  board)  means  one,  as,  1  dog,  1 

book.   When  writing,  we  make  it  thus :  — ^  CiOG^j   /    t'OOri, 

Note. — An  oral  lesson  like  the  following  may  precede  Lesson  II. 

Oral  Lesson. — "  How  many  pencils  have  I  in  my  right  hand?" 
Ans. — "  One  pencil."  "  How  many  in  my  left  hand?"  Ans. — 
"One  pencil."  " How  many  in  both ?"  Ans. — " Two  pencils." 
"  How  many  pieces  of  chalk  have  I  in  my  right  hand?"  Ans. — 
"One."  "In  my  left?"  ^wj.  —  "One."  "In  both?"  Ans.— 
"  Two."     "  How  many  fingers  do  I  hold  up  ?"     Ans. — "  Two." 

"  One  pen  and  one  pen  are,  how  many  pens  ?"  "  If  I  lay  down 
one  pen,  how  many  shall  I  have  left  ?"  Lay  down  one  pen,  and 
show  the  remaining  one.  "  How  many  more  must  I  get  to  have 
two  ?"  Taking  two  pens  in  the  right  hand  and  one  in  the  left, 
ask :  "  How  many  pens  have  I  in  my  right  hand  ?"  "  How  many 
in  my  left  hand?"  " How  many  more  in  my  right  hand  than  in 
my  left?"  "How  many  less  in  my  left  hand  than  in  my  right 
hand?"  "If  I  should  pass  one  from  my  right  hand  to  my  left 
hand,  how  many  would  there  be  in  my  right  hand?"  "How 
many  would  there  be  in  my  left?" 

Ask  such  questions  as  these,  illustrating  each  by  familiar 
objects,  and  continuing  the  exercise,  till  the  numbers  two,  and 
one,  and  their  relations  to  each  other,  are  perfectly  understood. 


LESSON    SECOND. 


LESSON  II. 


Two  boys.     2  boys.      S  ^ayd 

Two  soap-bubbles.  2  soap-bubbles.  S  i^ooA.'^OMed. 

Two.     2.     S.     II. 

The  mark  2,  or  2 j  is  called  i]iQ  figure  two. 

How  many  boys  do  you  see  in  the  picture  ? 
How  many  soap-bubbles  do  you  see  in  the  picture  ? 

A.  To  THE  Teacher.  —  The  following  questions  should  first 
be  asked  by  substituting  concrete  in  place  of  the  abstract  num- 
bers. Thus :  "  How  many  apples  are  1  apple  and  1  apple  ?  1 
pear  and  how  many  pears  are  two  pears?"  &c.  The  pupils 
should  be  taught  to  make  such  changes  for  themselves.  The  work 
is  not,  however,  mastered  till  the  abstract  numbers  and  operations 
are  mastered. 

1.  How  many  are  1  and  1  ? 

2.  1  and  how  many  are  2  ? 

3.  1  from  2  leaves  how  many  ? 

4.  How  many  must  be  taken  from  2  to  leave  1  ? 

5.  How  many  more  are  2  than  1  ? 


10  COLB  urn's    first    PART. 


B.  1.  George  blew  1  soap-bubble,  and  Joseph 
blew  1.    How  many  did  both  blow? 

2.  There  were  2  soap-bubbles  in  the  air,  but  1  of 
them  burst.     How  many  remained  ? 

3.  Sarah  has  2  dolls,  and  Mary  has  1.  How 
many  more  has  Sarah  than  Mary? 

4.  A  Story  about  James.  —  James  was  a  little 
boy  who  lived  in  the  country,  and  studied  the  First 
Book  of  Arithmetic.  On  his  way  to  school  one  day, 
he  found  2  apples.  At  recess,  he  gave  1  of  them  to 
his  Teacher,  and  ate  1 ;  but  just  before  recess  was 
over,  he  received  a  present  of  1  from  a  schoolmate. 
After  school,  he  found  1  under  a  tree,  and  gave  1 
to  a  little  boy  whom  he  met.  When  he  reached 
home,  he  roasted  all  he  had  left.  How  many  did  he 
roast  ? 


LESSON  III. 


Three  rabbits.     3  rabbits.     3  ^a^'^cl 

Three.     3.     J.   III. 

The  mark  3,  or  3 ^  is  called  the  figure  three. 


LESSON    THIRD.  11 


To  THE  Teacher. — Oral  lessons,  like  those  in  Lessons  I.  and  II., 
should  be  continued  in  this  and  the  subsequent  lessons.  They 
will,  better  than  any  lesson  from  the  book,  and  better  than  any 
mere  description,  lead  the  pupil  to  comprehend  the  nature  of 
numbers,  and  numerical  operations. 

A.  1.  How  many  are  2  and  1  ? 

2.  How  many  are  1  and  2? 

3.  How  many  are  1  and  1  and  1  ? 

4.  2  and  how  many  are  3  ? 

5.  1  and  how  many  are  3  ? 

6.  2  from  3  leaves  how  many? 

7.  1  from  3  leaves  how  many? 

8.  How  many  more  are  3  than  2  ? 

9.  How  many  more  are  3  than  1? 

B.  1.  Edward  had  2  tame  rabbits,  and  his  cou- 
sin gave  him  1  more.     How  many  had  he  then  ? 

Solution. — If  Edward  had  2  tame  rabbits,  and  his  cousin  gave  him 
1  more,  he  would  then  have  2  rabbits  and  1  rabbit,  which  are  3 
rabbits. 

Solution  2d. — The  2  rabbits  which  he  had,  and  the  1  rabbit  which 
his  cousin  gave  him,  would  make  2  rabbits  and  1  rabbit,  which  are  3 
rabbits. 

Note. — Such  reasoning  processes  as  the  foregoing  are  of  great 
value ;  for  they  teach  children  how  to  trace  the  connection  be- 
tween the  problems  and  the  numerical  operations,  and  thus  how 
to  reason;  and  they  prepare  the  way  for  the  solution  of  more 
complicated  problems.  A  little  attention  to  them  now,  will  save 
much  labor  both  to  teacher  and  pupil  in  the  higher  departments 
of  Arithmetic. 

2.  A  cross  dog  afterwards  killed  2  of  Edward's 
rabbits.     How  many  had  he  left  ? 

3.  Emma  had  3  rabbits,  1  of  them  was  black,  and 
the  rest  were  white.     How  many  were  white  ? 


12  colburn's   first  part. 


4.  A  Story  about  Carrie. — Carrie  was  a  bright- 
eyed  little  girl  who  lived  in  a  village.  One  day  she 
cut  out  2  paper  dolls,  and  the  next  day  she  cut  out 
1  more.  She  then  gave  1  to  her  playmate,  Martha, 
who  came  to  see  her,  and  1  to  Maria.  She  after- 
wards cut  out  2  more,  but  through  carelessness,  let 
1  fall  into  the  fire,  when  her  mother  cut  out  1  very 
nice  one,  and  gave  it  to  her.  How  many  had  she 
then  ? 

To  THE  Teacher. — Make  additional  problems,  and  encourage 
the  pupils  to  do  it  for  themselves,  arranging  them  somewhat  in 
the  form  of  stories,  to  increase  their  interest.  One  problem  pro- 
posed by  a  pupil,  and  solved  by  a  class,  will  be  of  more  value  in 
an  educational  view  than  many  proposed  by  a  teacher  or  author. 


LESSON   IV. 


Four  reapers.     4  reapers.     A  lea^ieid. 

Four.     4.      A.     IV. 

The  mark  4,  or  Aj  is  called  t\\(i  figure  four. 

Note. — It  should  be  made  a  part  of  the  lesson  for  the  pupil  to 
write  out  the  exercises  in  abstract  numbers.     He  will  thus  learn 


LESSON    FOURTH.  13 


to  use  figures  and  mathematical  signs,  and  to  write  out  arithme- 
tical work  neatly  and  correctly. 

Explanation. — A  cross  made  thus,  -\-,  is  sometimes  used  in  place 
of  "and"  in  such  questions  as,  how  many  are  1  and  2?  In  like  man- 
ner, "2  -|-  2  are  4"  mean  the  same  as  "  2  and  2  are  4."  This  sign  is 
also  called  ;;?m«,  and  sometimes  the  sign  of  addition. 

A.  1.  3+1?   3.  2+2?     5.  1+1+2? 
2.  1  +  3?   4.  1  +  2  +  1?   6.  2+1  +  1? 

Explanation.  —  Read  and  perform  the  following  questions,  and 
similar  ones  throughout  the  book,  as  though  the  words  "  how  many" 
were  put  in  the  place  of  the  star.  Thus,  the  question  "2  +  *  =  4?'' 
means  the  same  as  "2  and  how  many  are  4?" 

B.  1.    2  +  *are4? 

2.  l  +  *are4? 

3.  3  +  -are4? 

C.  1.  1  from  4? 
2.  2  from  4  ? 

D.  How  many  more  are — 
L  4  than  3? 
2.  4  than  1?  4.    3  than  1? 

E.  1.  George  has  2  apples,  and  Rufus  has  1. 
How  many  have  both  ?  How  many  more  has  George 
than  Rufus  ? 

2.  Edward  had  2  marbles,  and  his  father  gave 
him  a  cent,  with  which  he  bought  2  more.  How 
many  had  he  then  ? 

3.  He  afterwards  lost  1,  and  gave  away  2.  How 
many  had  he  left  ? 

4.  Jane  had  1  picture-book,  and  on  her  birth-day 
her  father  gave  her  1  more ;  her  mother  gave  her  1, 
and  her  uncle  Henry  sent  her  1,  which  was  very 
pretty.     How  many  had  she  then  ? 


4. 
5. 
6. 
3. 
4. 

l  +  l  +  *are4? 
l  +  2  +  *are4? 
2+1  +  *  are  4? 
3  from  4  ? 
1  from  3  ? 

3. 

4  than  2? 

14 


COLBURN    S    FIRST     PART. 


5.  There  were  3  robins  on  a  cherry-tree,  but  1  of 
them  flew  away,  and  2  others  came  to  the  tree.  A 
naughty  boy  throw  a  stone  to  knock  down  some 
cherries,  which  so  frightened  the  robins  that  4  of 
them  flew  away.  How  many  robins  were  left  on  the 
tree? 


LESSON  V. 


Five  toy-horses.     5  toy-horses.     5  ^^-no^^ed. 

Five.     5.    5.    V. 

The  mark  5,  or  6 j  is  called  the  figure  five. 

A.  1.     4  +  1?      4.     2  +  3?  7.     2  +  1  +  2? 

2.  1+4?      5.     3  +  1  +  1?       8.     1  +  1  +  3? 

3.  3  +  2?       6.     1  +  2  +  2?       9.     2  +  2  +  1? 

B.  ExPLAXATiON.  —  Two  parallel  lines  drawn  thus,  =,  form  what 
is  called  the  sign  of  eqxuility,  which  is  often  used  in  place  of  "  are"  in 
such  cases  as  "2+2  are  4,"  which  would  then  be  written  "  2  +  2  =  4." 
This  may  be  read  "2  and  2  are  4/'  or  "2  plus  2  are  4/*  or  "2  plus  2 
equal  4." 

1.  2+*=5?  4. 

2.  l  +  *=5?  6. 

3.  2  +  *=5?  6. 


2+l+*=5? 
2+2+*=5? 


LESSONFIFTH.  15 


c. 

1.     2  from  5? 

3. 

1  from  5  ? 

2.     4  from  5  ? 

4. 

3  from  5  ? 

D. 

How  many  more 

are  — 

- 

1.     5  than  2? 

3. 

5  than  3  ? 

2.     5  than  4? 

4. 

5  than  1  ? 

E.  1.  Edwin  had  2  cents,  but  he  afterwards  found 
3,  and  spent  4.     How  many  had  he  left  ? 

2.  Arthur  had  a  half-dime,  which,  as  you  know, 
is  worth  just  5  cents.  He  went  to  a  store  and  bought 
some  nuts  for  2  cents,  and  some  candy  for  1  cent, 
giving  in  payment  his  half- dime.  How  many  cents 
ought  he  to  receive  back  ? 

3.  Mr.  French  had  3  black  horses,  2  white  horses, 
and  1  grey  horse.  He  sold  his  grey  horse,  and  1 
of  his  black  ones.     How  many  had  he  left  ? 

4.  Near  a  village  lived  a  poor  w^oman  named  Lucy, 
but  everybody  called  her  Aunt  Lucy.  In  the  sum- 
mer she  would  pick  blackberries  to  sell.  One  day 
she  picked  3  quarts  in  one  pasture,  and  1  in  another, 
when,  meeting  a  gentleman  from  the  village,  she  sold 
him  2  quarts.  She  picked  3  quarts  more,  and  started 
to  go  home.  On  her  way,  she  sold  2  quarts  to  one 
man,  and  1  quart  to  another.  How  many  had  she 
left  ? 


16 

colburn's  first   part. 

LESSON  VI 

.^VX-^^^1 

^  1 

L 

^^  -  ^!^?^^^^ 

^ 

t. 

r^^mMky^y^WwoBjK 

j^^^i 

jr^^ffiM^^Pg|jB|«fflB|B 

^ 

Six  birds.     6  birds. 

^   vek(/<i- 

Six.     6.      6.     VL 

The  mark  6,  or  ^^  is  called  the  figure  six. 

A. 

1.    5  +  1?        4.    2+4?         6.    2  +  2  +  2? 

2.    1  +  5?         5.    3  +  3?         7.    1  +  3  +  2? 

3.    4  +  2? 

8.    1  +  2+2? 

B. 

1.    2  +  *=G?          4. 

l+l+l+*=6? 

2.    3  +  *=6?           5. 

l+2+*=6? 

3.    4  +  *=6?          6. 

l+l  +  2  +  *6? 

C. 

1.    4  from  6  ?          4. 

3  from  6  ? 

2.    1  from  6?          6. 

5  from  6  ? 

3.    2  from  6?          6. 

6  from  6  ? 

D. 

Explanation.  —  6  less  2  means  6  diminished  by  2,  or  made 

smalle 

r  by  2,  which  is  just  the  same 

as  "  2  from  6."     Hence  6  less 

2=4; 

5  less  3  =  2,  <fcc. 

1.     6  less  2?       3.    6  less  3  ?       5.    4  less  2  ? 

2.    5  less  3?       4.    5  less  1  ?       6.    6  less  4? 

E. 

1.  There  were  6  boys  at  play  ;  2  of  them  were 

LESSON    SEVENTH. 


17 


flying  their  kites,  and  the  rest  were   rolling   their 
hoops.     How  many  were  rolling  their  hoops? 

2.  A  pedler  had  6  plaster  birds  on  a  tray.  2  of 
them  were  painted  yellow,  1  of  them  was  painted  red 
and  brown,  and  the  rest  were  painted  red  and  black. 
How  many  were  painted  red  and  black  ? 

3.  A  hunter  shot  1  partridge,  3  quails,  and  2 
pigeons:     How  many  birds  did  he  shoot  in  all  ? 

4.  Julia  picked  3  white  roses,  and  3  red  ones. 
How  many  did  she  pick  in  all  ?  She  gave  2  red 
roses  and  1  white  rose  to  her  teacher,  and  1  white 
rose  to  her  friend  Lydia.     How  many  had  she  ? 


LESSON   VII. 


Seven  hens.     7  hens.      ^  nend* 
Seven.     7.     /.    VII. 
The  mark  7,  or  7^is  called  the  fiijfure  seven. 
1.     6+1?         4.     2  +  5?  7.     3+2+2? 


2. 
3. 


1  +  6? 
5+2? 


5. 
6. 


2* 


3+4? 

4  +  3? 


9. 


1  +  1  +  5? 

2+1+4? 


18 


COLBURN    S    FIRST    PART. 


B.  1. 

2. 
3. 

2+*=7? 
2+3+*=7? 

C.  1. 

2. 

6  from  7  ?     3. 
4  from  7  ?     4. 

4.  2  +  l  +  *-7? 

5.  l  +  2  +  l  +  *=7? 

6.  2  +  2  +  2+*=7? 

2  from  7?     5.    1  from  7? 

3  from  7?     6.    5  from  7  ? 

D.  1.     7  less  3  ?      3.    7  less  2  ?      5.    7  less  5? 
2.     7  less  1?      4.    7  less  6?     6.    7  less  4? 

E.  1.  A  farmer  had  seven  grej  liens.  He  sold  2 
of  them,  and  a  fox  killed  1  of  them.  How  many  did 
he  hnve  left  ?  If  he  should  afterwards  sell  2  more, 
and  huy  4  small  white  hens,  how  many  would  he 
then  have? 

2.  Alfred  had  a  half-dime  and  4  €ents  ;  hut  he 
exchanged  tlie  half  dime  fur  its  value  in  cents.  How 
many  cents  did  he  then  have  ?  He  was  so  unfortu- 
nate as  to  lose  3  of  his  cents,  and  he  gave  3  more 
for  a  three-cent  piece.  How  many  cents  had  he  then 
left? 

3.  6  boys  were  at  play  together ;  1  of  them  got 
hurt,  and  went  home,  and  2  were  called  a\^ay  by 
their  friends  ;  but  very  soon  4  more  boys  cm  me  out 
to  play.  These  played  together  till  all  but  3  got 
tired,  and  sat  down  to  rest.  How  many  sat  down 
to  rest  ? 


LESSON     EIGHTH. 


19 


LESSON  VIII. 


Eight  persons.     8  persons.     8 ^ZdontX. 

Eight.     8.     S.    VIII.  , 

The  mark  8,  or  8 j  is  called  \he  figure  eight. 


A.  1.  7  +  1? 

2.  1+7? 

3.  6+2? 

B.  1.  2+*=8? 

2.  4+*=8? 

3.  8+*=8? 

C.  1.  7  from  8? 

2.  2  from  8  ? 

3.  4  from  8  ? 


4.  2  +  6? 

5.  5  +  3? 


6.  3  +  5? 

7.  4+4? 

8.  2  +  2+2  +  2? 


4.  3+2+*=8? 

5.  l  +  l+2+l  +  *==8? 

6.  •  l  +  2  +  3  +  *=8? 

4.  (5  from  8? 

5.  3  from  8  ? 

6.  5  from  8  ? 


20  COLBURN*S    FIRST    PART. 


D.  llow  many  more  are  — 

1.  8  than  5?  ^  8  than  3? 

2.  8  than  1?  5.  8  than  2? 

3.  8  than  6  ?  6.  8  than  4  ? 

E.  Explanation.  —  A  single  mark  made  like  a  dash,  thus,  — ,  is 
often  used  in  place  of  the  -word  "  less."  For  instance :  "8  —  3  =  6" 
means  the  same  as  "  8  less  3  =  5." 

1.  8—3?        3.     8—5?        6.     8—6? 

2.  8—7?        4.     8—2?        6.     8—4? 

F.  1.  In  a  ferry-boat  were  4  ferrymen,  2  ladies, 
and  2  gentlemen.  How  many  persons  were  in  the 
boat? 

2.  A  farmer  had  8  little  pigs.  He  sold  2  to 
one  man,  and  2  to  another.    How  many  had  he  left  ? 

3.  Sarah's  mother  gave  her  3  dresses  for  her  doll, 
her  sister  Susan  gave  her  2,  and  her  aunt  Mary  gave 
her  enough  to  make  up  8.  How  many  did  her  aunt 
Mary  give  her  ? 

4.  Alfred  found  3  chestnuts  under  one  tree,  4 
under  another,  and  1  under  another.  He  soon  after 
ate  2,  when,  having  the  ill  luck  to  fall,  he  lost  3. 
He  afterwards  found  4  more,  when,  seeing  a  pretty 
squirrel  run  into  a  hole  in  a  tree,  he  put  in  3  chest- 
nuts for  the  squirrel  to  eat.  How  many  chestnuts 
had  he  left? 


LESSON     NINTH. 


21 


LESSON  IX. 

,il-s^'-,lir1:ri!«rtj 


<j^/^-: 


Nine  ducklings.     9  ducklings,     p  cUi,cn/cna<i: 

Nine,      9.     p.     IX. 

The  mark  9,  or  P ^  is  called  the  figure  nine. 


A.  1. 

2. 
3. 
4. 

B.  1. 

2. 
3. 
4. 

C.  1. 

2. 

3. 


8  +  1? 

1  +  8? 

7  +  2? 

2  +  7? 

5+*=9? 
2+*=9? 
4+*=9? 
3+*=9? 

8  from  9  ? 
6  from  9  ? 

3  from  9  ? 


6+3? 

3  +  6? 
5+4? 

4  +  5? 

5. 
6. 

7. 
8, 


9. 
10. 
11. 
12. 


3+1+3? 
2+2+5? 
3+2+3? 
1+2+4? 

:9? 


2  +  5  + 
3+5+*=9? 
3+3+*=9? 
2+2+2+*=9? 


4. 
5. 
6. 


7  from  9  ? 
4  from  9  ? 
2  from  9  ? 


22  coLB urn's  first  part. 


D.  1.     9—5?  4.     9—4—2? 

2.  (?— 7?  5.     9—2—3? 

3.  9— 1>?  6.     9—5—2? 

E.  1.  If  a  boy  should  have  6  cents,  and  receive 
a  present  of  2  more,  how  many  would  he  have  ?  If 
he  should  spend  3,  and  then  have  4  given  him,  how 
many  would  he  have  ? 

2.  Daniel  had  3  baskets.  The  first  was  a  red 
one,  which  held  3  quarts ;  the  second  was  a  blue  one, 
which  held  4  quarts  ;  the  third  was  a  yellow  one, 
which  held  2  quarts.  How  many  quarts  would  all 
hold? 

3.  One  day  he  picked  so  many  berries,  that  he 
filled  them  all ;  but  he  sold  what  there  was  in  the 
blue  basket.     How  many  quarts  had  he  left  ? 

4.  He  emptied  the  contents  of  his  red  basket  into 
his  blue  basket.  How  many  more  quarts  would  it 
take  to  fill  it? 

5.  David  had  3  marbles,  and  Austin  had  4;  but 
David  found  4,  and  Austin  lost  2.  Thej  then  agreed 
to  put  what  they  had  into  a  littie  box.  How  many 
marbles  did  they  put  into  the  box  ? 


LESSON    TENTH. 


23 


LESSON  X. 


Ten  herrings.    10  herrings.     ^0  nezuna^. 

Ten.      10.     ^0.      X.  ■ 

The  mark  0  is  called  ihejlgure  naught,  or  zero. 


A.  1.     9  +  1? 

2.  1+9? 

3.  8  +  2? 

4.  2  +  8? 

5.  7  +  3? 
3  +  7? 
6+4? 


6. 

7. 


1.  2+*=10? 

2.  3+*==10? 

3.  5  +  *=10? 

4.  4+*=10? 


8.  4+6? 

9.  5+5? 

10.  4  +  3  +  2? 

11.  2  +  3  +  5? 

12.  1+2  +  2  +  2  +  2? 

13.  2+2+2+2+2? 

6.  l+4+2+*=10? 

6.  3+l  +  2+*=10? 

7.  2  +  4+3  +  *=10? 

8.  4  +  l  +  2  +  *=10? 


24  colburn's  first  part. 


(J.  How  many  more  are  — 

1.  10  than  6?         4.  7  than  4? 

2.  8  than  3?  5.  10  than  7? 

3.  10  than  5  ?  6.  10  than  8  ? 

D.  1.     10—8?  5.     10—5+3? 

2.  10—7?  6.     10—4—3  +  6? 

3.  10—6?  7.     10—3  +  2—4? 
6.     10—3?  8.     10—4—4+5? 

E.  1.  Ahunter  shot  3Hrds  from  oneflock,  2fiom 
another,  and  5  from  another.  How  many  did  he 
shoot  in  all  ? 

2.  Anna  says  she  has  10  picture-books,  of  which 
her  mother  gave  her  3,  her  teacher  gave  her  1,  her 
aunt  gave  her  1,  her  uncle  gave  her  1,  and  her  fa- 
ther gave  her  the  rest.  How  many  did  her  father 
give  her  ? 

3.  Edward  had  a  half-dime,  a  three-cent  piece, 
and  2  cents.  How  many  cents  were  they  worth? 
He  bought  an  apple  for  2  cents,  an  orange  for  3 
cents,  some  candy  for  1  cent,  and  some  raisins  for  3 
cents.    How  many  cents  had  he  left  ? 

4.  Albert  and  Timothy  went  a-fishing  one  day. 
Albert  caught  3  perch,  2  pickerel,  and  4  trout. 
Timothy  caught  2  perch,  3  pickerel,  3  trout,  and  2 
eels.  How  many  more  fish  did  Timothy  catch  than 
Albert  ?  Albert  gave  his  4  trout  in  exchange  for 
Timothy's  2  perch  and  3  pickerel.  How  many  fish 
had  each  boy  then  ? 


LESSON    ELEVENTH. 


25 


LESSON  XI. 


Eleven  arrows.     11  arrows. 
Eleven.     11.     //.    XI. 


/'/  aao-iefj^. 


A. 

1. 

10  +  1? 

8. 

6+5? 

2. 

9+2? 

9. 

5+6? 

3. 

2+9? 

10. 

4+2+3? 

4. 

8  +  3? 

11. 

2+5+4? 

5. 

3+8? 

12. 

2+2+1+2+2+2? 

6. 

7+4? 

13. 

1+2+2+2+1+2? 

7. 

4+7? 

14. 

3+1+3+2+2? 

B.  1.  5+*=ll? 

2.  2  +  *=ll? 

3.  4  +  *=ll? 

C.  1.  *  from  11  =  5? 

2.  *  from  11=6? 

3.  *  from  11  =  7? 


4.  2+2+5  +  *=ll? 

5.  3+4  +  *=ll? 

6.  3  +  3  +  3+*=ll? 


4. 
5. 
6. 


*  from  11=4? 

*  from  11  =  8? 

*  from  11  =  3  ? 


26  COL  burn's  rmsT   part. 


D.  1.  11—4?  8  11—3? 

2.  11—2?  9.  11—10? 

3.  11—6?  10.  5+4  +  2-6? 

4.  11—8?  11.  1  +  2+4  +  4—3? 

5.  11—5?  12.  2  +  5  +  3—8? 

6.  11—9?  13.  3  +  1  +  5—3? 

7.  11—7?  14.  7  +  4—3—3? 

E.  1.  A  person  was  shooting  arrows  at  a  target, 
and  I  observed  that  when  he  had  shot  3  arrows,  and 
placed  another  in  his  bow,  there  were  7  lying  on  the 
ground.     How  many  were  there  in  all  ? 

2.  William  owned  3  arrows,  George  owned  2,  and 
Rufus  5.  How  many  did  they  all  own  ?  One  after- 
noon, as  they  were  playing  with  their  bows  and 
arrows,  William  lost  1  arrow,  Rufus  lost  1  and  broke 
1,  and  George  found  a  very  nice  one,  which  some  boy 
had  lost.     How  many  arrows  had  the  boys  then  ? 

3.  One  beautiful  afternoon  in  June,  Emma  and 
Hannah  went  out  to  gather  wild  flow^ers,  and  make 
boquets.  Emma  made  4,  and  Hannah  made  5,  when 
they  put  the  rest  of  their  flowers  together,  and  made 
2  very  pretty  boquets.  They  put  them  all  in  a  bas- 
ket, and  went  home.  They  gave  3  to  Hannah's 
mother,  and  1  to  her  sister ;  and  they  gave  3  to 
Emma's  aunt,  and  2  to  her  teacher  ;  after  which, 
Emma  took  1,  and  Hannah  took  the  rest.  How 
many  did  Hannah  take? 


LESSON    TWELFTH. 


27 


LESSON  ZII. 


Twelve  eggs.     12  eggs.     /J* 
Twelve.    12.    /J*.    XII. 


1. 

2. 
3. 
4. 

5. 
6. 

7. 


10  +  2? 

2  +  10? 
9+3? 

3  +  9? 

8+4? 

4  +  8? 
7+5? 


B.  1.  4+*=12? 

2.  5  +  *=12? 

3.  3  +  *=12? 

0.  1.  6  from  12  ? 

2.  9  from  12? 

3.  2  from  12? 


9. 
10. 
11. 
12. 
13. 
14. 


5+7? 
6  +  6? 

3+3+3+3? 
3+2+3+2? 

1+4+3+3? 
2+5+2+3? 
4+2+2+4? 


4.  3+4+3  +  *==12? 

5.  2  +  l  +  4  +  *  +  12? 

6.  2  +  l+2+*=12? 

4.  8  from  12? 

5.  10  from  12? 

6.  7  from  12  ? 


28  colburn's   first   part. 


D.  1.  12—4?  6.  12—9? 

2.  12—3?  7.  12—3—4? 

3.  12—7?  8.  2  +  7  +  3—6? 

4.  12—8?  9.  12—5—2? 

5.  12—10?  10.  1  +  8  +  2—4? 

E.  1.  Alfred  found  a  hen's  nest,  with  a  large 
number  of  eggs  in  it.  He  took  out  3,  and  then  took 
out  4  more,  when  he  found  that  there  were  5  left  in 
the  nest.  How  many  were  there  in  the  nest  at  first  ? 
When  he  was  putting  the  eggs  back,  he  carelessly 
broke  2.     How  many  were  then  left  ? 

2.  Alice  had  11  little  chickens,  but  2  of  them 
died,  2  of  them  got  lost,  and  a  rat  killed  3.  How 
many  then  remained  ? 

3.  Benjamin  earned  a  half-dime,  and  found  a 
three-cent  piece  and  4  cents.  He  bought  a  top  for 
8  cents,  after  which  he  received  a  present  of  8  cents. 
How  much  money  had  he  then  ? 

4.  Annie  had  7  pictures,  and  Emma  had  5.  Annie 
gave  away  3  pictures,  and  Emma  received  a  present 
of  2  ;  after  which  Emma  lost  3,  and  Annie  found  1. 
How  many  pictures  had  each  girl  then  ?  How  many 
had  both  ? 


LESSON 

IHIRTEENTH.                    29 

LESSON  XIII 

« 

/f"^*^^/^^^^ 
^W^          '^^^#*I 

^ 

g 

||KMyfev^ 

1 

11^^ 

f 

Chirteen  sheep. 

18  sheep.    /J*  ^/^^. 

Thirteen.      13. 

/3. 

XIII. 

A. 

1. 

10  +  3? 

7. 

7+6? 

2. 

3  +  10? 

8. 

6+7? 

3. 

9+4? 

9. 

2+3+4+3? 

4. 

4+9? 

10. 

1+4+2+6? 

5. 

8  +  5? 

11. 

4+3+3+4 

6. 

5  +  8? 

12. 

•1  +  5  +  3+4? 

B. 

1. 

3+*=13? 

4. 

3  +  4  +  *=13? 

2. 

5  +  *=13? 

5. 

4  +  2  +  4+*=13? 

3. 

6  +  *=13? 

6. 

2  +  3+4  +  *=13? 

C. 

1. 

9  from  13  ? 

4. 

6  from  13? 

2. 

7  from  13  ? 

6. 

5  from  13  ? 

3. 

4  from  13  ? 

6. 

8  from  13  ? 

3* 

H: 


80  colburn's  first  part. 

D.  1.     13—*=7?         4.    13— *=9? 

2.  13— *=4?         5.    leS— *=6? 

3.  13— *=8?  6.     13— *=5? 

E.  1.  Mr.  Green  owns  13  sheep,  and  Mr.  Allen 
owns  7.  How  many  more  does  Mr.  Green  own  than 
Mr.  Allen  ?  If  Mr.  Green  should  sell  4  sheep  to 
Mr.  Allen,  how  many  would  each  have  ? 

2.  A  farmer  had  3  sheep  in  one  pasture,  5  in 
another,  and  4  in  another ;  but  at  night  he  drove 
them  all  into  one  pen.  How  many  were  there  in 
the  pen  ?  The  next  day  he  drove  6  of  them  into 
one  pasture,  2  into  another,  and  the  rest  into  an- 
other.  How  many  did  he  drive  into  the  last  pasture  ? 

3.  A  little  boy  had  13  marbles.  He  lost  4,  and 
gave  away  3,  when,  finding  it  was  school-time,  he 
put  the  rest  into  a  box.  When  he  came  from 
school,  he  found  his  little  brother  Erastus  had  been 
playing  with  the  box,  and  had  lost  3  of  the  mar- 
bles. How  many  were  left  in  the  box  ?  His  father 
afterwards  gave  him  2  cents,  with  which  he  pur- 
chased 5  marbles.  How  many  marbles  had  he 
then  ? 


LESSON 

FOURTEENTH.                      31 

LESSON 

XIV. 

i 

I 

^^t^^saw^^^^^^^C^ 

^^p^^s 

^^^^^m 

^j^^^K 

S^ 

Fourteen  barrels 

3.     14  barrels.     /^  ^atte/d. 

Fourteen.     14. 

/A. 

XIV. 

A. 

1. 

10+4? 

6. 

6  +  8? 

2. 

4  +  10? 

7. 

7  +  7? 

3. 

9  +  5? 

8. 

2+1+7+4? 

4. 

5+9? 

9. 

4+2+8? 

5. 

8r6? 

10. 

3+4+2+4? 

B. 

1. 

7  +  *=14? 

4. 

3+4  +  *=14? 

2. 

5  +  *=14? 

6. 

4+6  +  *=14? 

3. 

6  +  *=14? 

6. 

2  +  4+3  +  *^14? 

C, 

1. 

9  from  14  ? 

4. 

8  from  14? 

2. 

6  from  14  ? 

5. 

7  from  14?  ,,,- 

i 

3. 

4  from  14  ? 

6. 

4  from  14?'      '^ 

32  colburn's  first   part. 


D.  1.     14—4—3?         4.     4  +  5  +  5—6  +  3? 

2.  14—6  +  3?         5.     8  +  2  +  3—5—3? 

3.  14—7+5?  6.     3  +  2+4  +  5—6—3? 

E.  1.  A  teamster  had  a  load  of  14  barrels.  He 
unloaded  6  at  the  store  of  Shaw  &  Co.,  3  at  a  railroad 
depot,  and  the  rest  at  the  store  of  Saunders  k 
Brown.  How  many  did  he  unload  at  the  last  place  ? 
Not  long  after,  he  had  a  load  of  13  barrels,  and  he 
unloaded  4  of  them  at  one  place,  and  3  at  another, 
after  which  he  took  on  his  truck  8  barrels  more. 
How  many  had  he  then  on  his  load  ? 

2.  Susan  has  6  books  without  pictures,  and  7  books 
with  pictures.     How  many  books  has  she  ? 

3.  Austin  has  14  books.  Waldo  had  6.  His  mo- 
ther gave  him  3,  and  his  father  gave  him  enough  to 
make  as  many  as  Austin.  How  many  did  his  father 
give  him  ? 

4.  One  day,  Henry  went  out  to  look  for  chest- 
nuts. He  found  6  under  one  tree,  3  under  another, 
and  5  under  another.  After  eating  8  of  them,  and 
finding  6  more,  he  went  home,  carrying  his  chest- 
nuts with  him.  He  gave  3  of  them  to  his  father,  3  to 
his  mother,  4  to  his  little  sister  Lucy,  and  the  rest  to 
his  brother  Francis.  How  many  did  he  give  to 
Francis  ? 


LESSON    FIFTEENTH. 


33 


LESSON   XV. 


Fifteen  apples.     15  apples.     ^5  a/?A^a 
Fifteen.     15.     <f5.    XV. 


A.  1.  10  +  5? 
2.  5+10? 

B.  1.  8  from  15  ? 

2.  6  from  15  ? 

3.  10  from  15? 

C.  1.  15-7  +  3? 

2.  15-6+4? 

3.  1,5-9-5? 

D.  1.  15-*=7? 

2.  15-*=6? 

3.  15-*=-9? 


3.  9  +  6? 

4.  6+9? 


5.  8+7? 

6.  7  +  8? 


4.  7  from  15? 

5.  5  from  15? 

6.  9  from  15? 

4.  8  +  2+5-7? 

5.  4+3+8-6? 

6.  2  +  7  +  5-8  +  3? 

4.  15-*=5? 

5.  15-*==8? 

6.  14-*=10? 


C 


34  colburn's  fibst  part. 


E.  1.  Mary  found  3  apples  under  one  tree,  4  apples 
under  another,  and  8  under  another.  How  many 
did  she  find  in  all  ?  As  she  was  bringing  them  to 
the  house,  she  stopped  to  play  with  her  kitten,  and 
accidentally  dropped  most  of  them,  as  you  see  in  the 
picture.  On  picking  them  up,  she  found  that  6  of 
them  were  bruised  a  little,  and  1  of  them,  which  the 
kitten  played  with,  was  bruised  very  badly.  The  rest 
were  not  bruised  at  all.  How  many  were  not  bruised 
at  all? 

2.  Julia  made  4  squares  of  blue  patch-work,  3 
squares  of  brown,  and  8  squares  of  red.  How  many 
did  she  make  in  all  ? 

3.  Hattie  hemmed  15  handkerchiefs,  5  of  them 
were  for  her  sister  Lydia,  2  were  for  her  brother 
Cyrus,  3  for  her  father,  and  the  rest  for  her  mother. 
How  many  did  she  hem  for  her  mother? 

4.  Augusta  received  4  merit-marks  on  Monday,  3 
on  Tuesday,  1  on  Wednesday,  2  on  Thursday,  and  5 
on  Friday,  and  on  Saturday  school  was  not  in  ses- 
sion. How  many  merit-marks  did  she  receive  through 
the  week?  How  many  more  than  Emeline,  who 
obtained  but  9  during  the  week  ? 


LESSON    SIXTEENTH. 


35 


LESSON  XVI. 


Sixteen  tents.      16  tents.     ^6  lent^. 
Sixteen.     16.     ^6.    XVI. 


A.  1.  10  +  6? 

2.  6  +  10? 

3.  9  +  7? 

B.  1.  8+*=16? 

2.  6  +  *=16? 

3.  2  +  5  +  *=16? 

C.  1.  16-6? 

2.  16-9? 

3.  16-7? 


4.    7+9? 


4.  3+4+2  +  *=16? 

5.  4+2+4  +  *=16? 
6      l+3+2+*=16? 

4  IG— *=-7? 

5  16-*=9? 

6.  16— *=8'^ 


30  colburn's  first  part. 


D.  1.  14=4+*?  6.  16=104-*? 

2.  16  =  6f*?  7.  13=10+*? 

3.  13=3+*?  8.  14=10  +  *? 

4.  12=2  +  *?  9.  12=10  +  *? 

5.  11=1  +  *?  10.  11=10  +  *? 

E.  1.     4  +  2?  6.     3  +  13? 

2.  14  +  2?  7.  1+4? 

3.  4  +  12?  8.  11+4? 

4.  3  +  3?  9.  1  +  14? 

5.  13  +  3? 

F.  1.  On  a  certain  muster-field,  tliere  were  8 
tents  in  one  row,  and  8  in  another.  How  many  were 
there  in  both  rows  ? 

2.  Albert  was  asked  how  many  chestnuts  he  had, 
to  which  he  replied,  "  If  I  should  give  my  father  3, 
my  mother  4,  my  little  sister  Anna  4,  and  my  bro- 
ther George  3,  T  should  have  but  2  left."  How  many 
chestnuts  had  he  ? 

3.  Lucy  read  6  pages  of  history  in  the  morning, 
and  10  in  the  afternoon,  but  when  questioned  about 
it,  she  found  that  she  had  forgotten  all  but  4  pages. 
How  many  had  she  forgotten  ? 

4.  I  had  3  dollars,  and  received  6  dollars  of  one 
man,  3  of  another,  and  2  of  another,  when  I  paid 
away  8  dollars,  after  which  I  received  4  dollars. 
How  many  had  I  then  ? 


LESSON     SEVENTEENTH, 


37 


LESSON  XYII. 


Seventeen  birds.     17  birds.    ^7  ^irc/d-. 
Seventeen.     17.     //.    XVII. 


A.  1.  10+7? 

2.  7+10? 

3.  8+9? 

B.  1.  4+5+*=17? 

2.  3+7  +  *=17? 


4.  9  +  8? 

5.  3  +  5+9? 

6.  7+2+8? 

3.  4  +  3+*=17? 

4.  4  +  4  +  *=17? 


C.  1.     *  from  17=8  ?       3.     *  from  17=7  ? 
2.     *  from  17=10  ?     4.     =f:  from  17=9  ? 


D.  1.     3  +  1  +  3? 

2.  13+1  +  3? 

3.  3  +  11  +  3? 


4.  2  +  2  +  2? 

5.  12  +  2  +  2? 

6.  2  +  12  +  2? 


38  COLBURN*S     FIRST     PART. 


E.  1.  4-f*-7?  7.  3  from  7? 

2.  4  +  *=17?  8.  3  from  17? 

3.  14  +  *=17?  9.  13  from  17? 

4.  3  +  *=6?  10.  1  from  6? 

5.  3  +  *-16?  11.  1  from  16? 

6.  134->K=16?  12.  11  from  16? 

F.  1.  Jane  gave  6  cents  to  a  beggar-woman,  Lucy 
gave  3,  Sarah  gave  5,  and  Abbj  gave  3.  How  many 
(lid  all  give  her  ? 

2.  The  beggar-woman  spent  10  cents  for  bread, 
after  which  Julia  gave  her  4  cents,  Nancy  gave  her 
3  cents,  and  Susan  gave  her  2  cents.  How  many 
cents  had  she  then  ? 

3.  A  gardener  picked  8  roses  from  one  bush,  7 
from  another,  and  2  from  another.  How  many  did 
he  pick  in  all  ?  He  put  4  of  the  roses  in  one  boquet, 
5  in  another,  and  3  in  another,  and  the  rest  in 
another.     How  many  did  he  put  in  the  last  boquet  ? 

4.  Samuel  bought  a  quart  of  molasses  for  10 
cents,  and  then  had  6  cents  left.  How  many  cents 
had  he  at  first  ? 

5.  He  made  his  molasses  into  candy,  9  sticks  of 
which  he  sold  for  7  cents,  and  the  remaining  8  sticks 
he  sold  for  6  cents.  How  many  sticks  did  he  sell  ? 
How  many  cents  did  he  receive  for  it  ?  How  many 
more  cents  did  he  receive  for  his  candy  than  he  paid 
for  his  molasses  ? 


LESSON     E I G  H  T  K  E  N  T  U . 


39 


LESSON  XVIII. 


Eighteen  books.     18  books,     /o  M<m<S: 
Eighteen.     18.      /<?.    XVIII. 


A.  1.  10  +  8? 

2.  8  +  9? 

B.  1.  8  +  *=18? 

2.  9+*=18? 

3.  10+*=18? 

C.  1.  *  from  7=3? 

2.  *  from  17=3? 

3.  *  from  17=13? 


3,     9+9? 


4.  2  +  2  +  *=14? 

5.  12  +  2+*=14? 

6.  2+12  +  *=14? 

4.  *  from  8=4? 

5.  *  from  18=4  ? 

6.  *  from  18  =  14? 


40  COL  burn's   first   part. 

D.  1.     8—2—2—2?        5.     9  +  6  +  3—4? 

2.  18—2—2—2?       6.     9  +  3+4—2? 

3.  18—12—2—2?     7.     10  +  3  +  4—3? 

4.  i7_3_3  +  4?      8.     18—4—4  +  3  +  2? 

E.  1.  One  "  Fourth  of  July,"  Robert's  father 
gave  him  a  dime,  his  mother  gave  him  a  half-dime, 
and  his  uncle  gave  him  a  three-cent  piece  ;  but  he 
exchanged  them  all  for  their  value  in  cents.  How 
many  cents  did  he  receive  for  them  ? 

2.  He  paid  8  cents  for  a  bunch  of  crackers,  and 
4  cents  for  torpedoes,  and  the  rest  of  his  money  to 
see  some  animals,  which  were  exhibited  in  a  tent. 
How  many  cents  did  he  pay  to  see  the  animals  ? 

3.  Mr.  Gay  owns  a  garden,  a  pasture,  a  wood-lot, 
and  an  orchard.  His  garden  contains  2  acres,  his 
pasture  6  acres,  his  wood-lot  4  acres,  and  his  orchard 
enough  to  make  up  18  acres.  How  many  acres  does 
his  orchard  contain  ?  If  he  should  sell  his  wood-lot 
and  orchard,  how  many  acres  would  he  have  left  ? 

4.  I  had  17  dollars  this  morning,  but  I  have  since 
bought  a  hat  for  4  dollars,  and  a  pair  of  boots  for  6 
dollars.  I  have  also  received  11  dollars,  which  a 
friend  owed  me,  and  paid  a  debt  of  7  dollars.  How 
much  money  have  I  now  ? 


LESSON    NINETEENTH.                       41 

LESSON  XIX. 

-z^^-r-^'T?^ 

%^-^-T r-„ 

^^^^Hipp 

Wk^m 

^^^R3 

^^p 

Nineteen  wild  geese.  19  wild 

geese.  ^PwuUaffede. 

Nineteen.     19.      /^.    XIX. 

A. 

1.     10  +  9?               5. 

14  +  2  +  *=19? 

2.     9  +  10?               6. 

4+2  +  *=19? 

3.     10  +  *=19?       7. 

19     3     3  ? 

4.     9  +  *-19?    ■     8. 

19     3     13? 

B. 

Ho;y  many  more  are — 

1.     9  than  2  ?           4. 

8  than  3  ? 

2.     19  than  2?          5. 

18  than  3  ? 

3      19  than  12  ?       6. 

18  than  13  ? 

C. 

1.    4  +  3+4+5?      6. 

4  +  6  +  *=19? 

2.     8  +  3+*=19?     7. 

9  +  3  +  5— *=8? 

3.     5  +  7+4    8?     8. 

10+4  +  4— *=11? 

4.     8+6  +  6—7?      9. 

3+3+3+3+3+3? 

5.     6  +  4+4+4?     10. 

1+3+3+3+3+3? 

42  colburn's   first   part. 


D.  1.  One  day,  Edward  and  Susan  saw  a  flock  of 
wild  geese,  which  contained  just  19.  Some  sports- 
men shot  4  of  them,  which  so  frightened  the  rest, 
that  6  of  them  flew  towards  the  east,  and  the  re- 
mainder towards  the  west.  Another  party  of  hunt- 
ers seeing  those  which  were  flying  towards  the  west, 
shot  2  of  them,  when  the  rest  flew  towards  the  east, 
and  joined  that  part  of  the  flock  which  had  first 
flown  in  that  direction.  How  large  a  flock  was  there 
then  ? 

2.  Frank  has  money  enough  to  buy  a  pencil  for 
3  cents,  a  pen  for  6  cents,  some  ink  for  4  cents,  and 
an  inkstand  for  5  cents.    How  much  money  has  he  ? 

3.  Frank's  sister  has  money  enough  to  buy  a  pen, 
and  inkstand  like  Frank's,  and  5  cents  worth  of 
paper.     How  many  cents  has  she  ? 

4.  Walter  and  Reuben  had  each  12  cents.  But 
Walter  earned  5  cents  by  doing  an  errand,  and 
Reuben  spent  5  cents  for  confectionery.  How  many 
cents  had  each  of  the  boys  then  ?  How  many  more 
had  Walter  than  Reuben  ? 


LKSSON     TWENTIETH. 


43 


LESSON  XX. 


Twenty  soldiers.     20  soldiers.      SO  Mu/ieu 
Twenty.     20.    20.    XX. 


A.  1.     10+10? 
2.     10  +  *=20? 


3.  20=*  tens? 

4.  20—10  ? 


B.  1.  3+*=.10?  4.  1+*=:10? 

2.  3  +  *=20?  5.  l-l-*=20? 

3.  13+*=20?  6.  ll+*=20? 

C.  1.  3  +  5+7+4?  4.  2+1+4+9  +  4? 

2.  2+9+4  +  6?  5.  4  +  9+2  +  2  +  2? 

3.  6  +  7  +  3  +  5?  6.  2  +  4+4+4+4? 


44  colburn's   first   part. 


7.  2  +  2  +  2  +  2  +  24-2  +  2  +  2  +  2  +  2? 

8.  1  +  2  +  2  +  2  +  2  +  2  +  2  +  2+2  +  2? 

9.  3  +  3  +  3-13  +  3  +  3? 

10.  1  +  3  +  3  +  3  +  3  +  3  +  3? 

11.  2+3  +  3  +  3  +  3  +  3  +  3? 

12.  20—2—2—2—2—2—2—2—2  +  2  +  2  ? 

13.  19-2—2-2—2—2—2-2—2-2  ? 

14.  20—3—3—3—3—3—3  ? 

15.  19—3—3—3—3—3—3  ? 

16.  18—3—3—3—3—3—3  ? 

D.  1.  Little  Willie  had  20  toy-soldiers.  Replaced 
3  in  front  for  officers,  and  then  arranged  the  rest  in 
3  rows,  placing  7  in  the  first  row,  6  in  the  second, 
and  the  rest  in  the  third.  How  many  did  he  place 
in  the  third  row? 

2.  Arthur  was  telling  his  mother  about  the  boys 
who  went  to  his  school.  He  said  that  4  of  them  had 
neither  hoops  nor  kites,  that  6  had  kites  only,  that 
7  had  hoops  only,  and  that  3,  including  himself,  had 
both  hoops  and  kites,  and  this  comprised  all  the 
boys  in  the  school.  How  many  boys  were  there  in 
the  school? 

3.  Isaac  and  Francis  were  playing  ball  with  Au- 
gustus and  Reuben.  Isaac  batted  the  ball  11  times, 
and  Francis  batted  the  ball  9  times.  Augustus  batted 
it  10  times,  and  Reuben  7  times.  How  many  times 
did  Isaac  and  Francis  bat  it  ?  How  many  times  did 
Augustus  and  Reuben  bat  it  ?  Isaac  caught  the  ball 
9  times,  and  Francis  caught  it  7  times.  Augustus 
caught  it  9  times,  and  Reuben  caught  it  10  times. 


LESSON    TWENTIETH.  45 


How  many  more  times  did  Augustus  and  Reuben 
catch  it  than  Isaac  and  Francis  ? 

4.  Three  idle  boys,  Thomas,  Joseph,  and  Samuel, 
were  disputing  about  their  examples.  Samuel  said 
he  performed  6  examples  on  Monday,  3  on  Tuesday, 
and  5  on  Wednesday.  Joseph  said  he  performed  3 
on  Monday,  5  on  Tuesday,  and  6  on  Wednesday. 
Thomas  said  he  performed  6  on  Monday,  3  on  Tues- 
day, and  5  on  Wednesday.  Each  thought  he  had 
performed  more  than  either  of  the  others ;  so  they 
quarreled  about  it.  N6w,  can  you  tell  who  had  done 
the  most  ?  William,  who  was  an  industrious  boy, 
performed  20  examples  on  Monday.  How  many 
more  than  Samuel  did  he  perform  on  that  day  ? 
How  many  more  than  Joseph  ?  How  many  more 
than  Thomas  ?  How  many  more  than  each  of  the 
others  performed  in  three  days  ? 

5.  Susan  and  Mary  had  each  9  oranges.  Susan 
gave  5  of  hers  to  Mary,  and  Mary  ate  2.  They  then 
put  what  they  had  left  together,  intending  to  keep 
them  until  the  next  week;  but  before  that  time  4 
of  them  had  spoiled.  They  then  so  divided  the  good 
ones  among  them,  that  Mary  had  6.  How  many 
had  Susan  ? 

G.  Mr.  Wheelock  had  6  dollars,  but  he  has  since 
received  9  dollars,  spent  10  dollars  for  broad-cloth, 
received  an  old  debt  of  2  dollars,  found  2  dollars, 
received  7  dollars  for  work,  paid  8  dollars  for  a  bar- 
rel of  flour,  2  dollars  for  a  barrel  of  apples,  and  lost 
6  dollars.     How  many  dollars  has  he  now  ? 


46  colburn's   first   part. 


LESSON   XXI. 

2  tens = twenty,  and  is  wi'itten  20,  or  SO, 

3  tens = thirty,  and  is  written  30,  or  30. 

4  tens = forty,  and  is  written  40,  or  AO, 

5  tens = fifty,  and  is  written  50,  or  SO, 

6  tens=sixty,  and  is  written  60,  or  OO, 

7  tens= seventy,  and  is  written  70,  or  /^O 

8  tens = eighty,  and  is  written  80,  or  oO, 

9  tens^ninety,  and  is  written  90,  or  ^0, 

10  tens = one  hundred,  and  is  Avritten  100,  or  /OO. 

E.   How  many  tens^re  there — 

1.  In  50?  4.     In  70?  7.     In  90? 

2.  In  80?  6.     In  20?  8.     In  40? 

3.  In  30?  6.     In  60?  9.     In  100? 

C.  "What  number  is  equal  to  each  of  the  following  : 

1.  3  tens?  4.     4  tens?  7.     6  tens? 

2.  9  tens  ?  6.     8  tens  ?  8.     2  tens  ? 

3.  10  tens  ?  6.     7  tens  ?  9.     5  tens  ? 

D.  How  will  you  write  each  of  the  following  num- 
bers in  fio-ures  ? 


LESSON    TWENTY-FIRST.  47 


1.  Forty?  4.     Ninety?  7.     Thirty? 

2.  Eighty?  5.     Seventy?  8.     Sixty? 

3.  Twenty?  6.     Fifty?  9.     One  hundred  ? 

E.  1.     4  tens  +  5  tens  ?     Then  40  +  60  ? 

2.  4  tens  +  4  tens  ?     Then  40  +  40  ? 

3.  7  tens  +  3  tens  ?     Then  70  +  30  ? 

4.  6  tens  -|-  *  tens  =  9  tens  ?     Then  60  +  *  =:  90  ? 
6.  4  tens  -f  *  tens  =  10  tens  ?   Then  40  +  *  =  100  ? 

6.  2  tens  +  *  tens  =  6  tens  ?     Then  20  +  ^  =  60  ? 

7.  8  tens  —  5  tens  ?     Then  80  -—  50  ? 

8.  4  tens  —  3  tens  ?     Then  40  —  30  ? 

9.  10  tens  —  3  tens  ?  Then  100  —  30  ? 

F.  1.     404-30+20?  7.     30+30+40—50? 

2.  50+20+30?  8.  20+20+20+20+20—70? 

3.  20+30+30?  9.  100—20—20—20—20+50? 

4.  20+40+20?  10.  90—00—20+^=60? 
6.  40+20+30?  11.  20+30+40—^  =  70? 

6.     30+30+40?  12.     100— 30— 30— 20+)t  =  80? 

G.  1.  A  man  gave  20  cents  for  Harpers'  Magazine,  20  for  Put- 
nam's, and  50  for  the  North  American  Review.  How  many  cents 
did  he  pay  for  all  ? 

2.  A  provision-dealer  had  20  bushels  of  potatoes.  He  after- 
wards bought  20  bushels  of  one  man,  30  of  another,  and  30  of 
another,  and  then  sold  40  bushels  to  one  man,  and  30  to  another. 
How  many  bushels  had  he  left  ? 

8.  A  farmer,  who  owned  90  sheep,  kept  40  in  one  pasture,  30 
in  another,  and  the  rest  in  another.  How  many  did  he  keep  in 
the  last  pasture  ?     He  drove  10  sheep  from  the  second  pasture 


48  colbuiin's   first   part. 


into  the  first,  and  20  from  the  second  into  the  third.     How  many 
were  then  in  each  pasture  ? 

4.  Sarah  found  20  walnuts  under  one  tree,  and  30  under  an- 
othef,  while  Lydia  found  30  under  one  tree,  and  20  under  another. 
How  many  did  each  find  ?     How  many  did  both  find  ? 

5.  Sarah  gave  20  nuts  to  Jane,  and  Lydia  gave  her  10.  How 
many  did  both  give  her  ?  How  many  had  each  left  ?  How  many 
had  the  three  girls  ? 

6.  George  has  20  cents,  Williams  has  20,  and  Edward  has  20 
more  than  George  and  William  together.  How  many  has  Edward? 
How  many  have  all  the  boys?  They  agreed  to  put  their  money 
together  and  purchase  some  articles  with  it.  They  bought  some 
paper  for  20  cents,  some  pencils  for  20  cents,  and  some  pens  for 
10  cents,  and  a  pretty  story-book  for  the  rest  of  their  money. 
How  much  did  the  story-book  cost  ? 


LESSON  XXII. 

A.  A  unit  is  a  single  thing,  or  one. 

2  tens  -|-  1  unit  =  twenty-one  =  21. 
2  tens  -\-  2  units  =  twenty-two  =  22. 

2  tens  -|-  9  units  =  twenty-nine  =  29. 

3  tens  -f-  1  unit  =  thirty-one  =  31. 

8  tens  -|-  7  units  =  eighty-seven  =  87. 

9  tens  -f-  9  units  =  ninety-nine  =  99. 

B.  Count  from  twenty  to  one  hundred^  thus :  — 
Twenty-one,  twenty-two,  twenty-three,  &c.,  &c. 

What  is  the  value  of — 

1.  20-J-8?  3.     30-1-7?  5.     60 -f  9? 

2.  40  -I-  6  ?  4.     90  -I-  3  ?  6.     20  -}-  5  ? 


LESSON    TWENTY-SECOND,  49 

C.  How  many  tens  and  units  are  there  — 

1.  In  64?  4.     In  28?  7.     Li23? 

2.  In  87?  6.     In  60?  ^     8.     In  19  ? 

3.  In  73?  6.     In  07?  9.     In  91  ? 

Explanation. — The  figure  vrhich  represents  the  units  of  a  number 
is  called  the  units'  figure,  and  that  which  represents  the  tens  is 
called  the  tens*  figure. 

Point  out  the  tens'  figure,  and  also  the  units'  figure,  in  the 
numbers  written  under  letter  C. 

The  position,  or  place  occupied  by  the  unit's  figure,  is  called  the 
units*  place,  and  that  occupied  by  the  tens*  figure  is  called  the  tens' 
place.     The  tens'  place  is  always  just  at  the  left  of  the  units. 

A  period,  called  the  decimal  point,  is  often  used  to  aid  in  determin- 
ing the  place  of  figures ;  the  first  place  at  the  left  of  the  point  being 
the  units'  place,  the  second  at  the  left  being  the  tens'  place,  and  the 
third  the  hundreds'.  When  the  point  is  not  written,  it  is  understood 
to  belong  at  the  right  of  the  given  number,  thus  making  the  right- 
hand  figure  the  units'  figure. 

E.  Write  each  of  the  following  numbers  : — 

1.  Seventy-nine.        3.     Fifty-seven.        5.     Eighty-six. 

2.  Twenty-four.         4.     Sixty -nine.  6.     One  hundred. 

F.  1.       4  +  2  +  2?  9.     3+4+4f  =  9? 

2.  24  +  2  +  2?  10.     3  +  4+^  =  29? 

3.  84  +  2  +  2?  11.     3+24+^  =  29? 

4.  4  +  72  +  2  ?  12.     3  +  74  +  *  =  79  ? 
6.       6  +  3  +  2?               13.     2 +  4  + ^-  =  10? 

6.  65  +  3  +  2^?  14.     2  +  4  +  *  =  40? 

7.  95  +  3  +  2?  15.    2  +  4  +  ^  =  70? 

8.  85  +  3  +  2?  16.     2  +  4  +  ^  =  100? 

G.  1.     2  +  5  +  3  +  3  +  2  +  5+3  +  5+2+3? 
2.     2  +  7  +  1  +  2  +  7  +  1  +  2  +  7  +  1+2? 

6  .  —  p  = 


' 1 

GO 

COLB  urn's    first    PART. 

3. 

3  +  3-f4-f34-34-4-f3-|-3-f4  +  3? 

4. 

53  +  3_[-4-f3  +  3  +  4-f3-f3-f44.3? 

6. 

24  4-2  +  4  +  1  +  7  +  2  -I.  4  +  6  -f.  2  + 5? 

6. 

35  +  3  +  2+3  +  7  +  1  +  9  +  2  +  4  +  2? 

7. 

42  +  8  +  5  +  3  +  2  +  4  +  6  +  2+7+1? 

8. 

67  +  3+1  +  3  +  5  +  1  +  2  +  3  +  4+1? 

9. 

14+6  +  4+6+3  +  7  +  5+5  +  4+4? 

10. 

31  +  2  +  4+3  +  5  +  5+8  +  2  +  3  +  4? 

H.  1. 

9  —  3  —  4?                     4.     10  —  2  —  5? 

2. 

29  —  3  —  4  ?                     5.     40  —  2  —  5  ? 

3. 

99  —  3  —  4?                      6.     50  —  2  —  45? 

I.    1. 

70  —  4  —  3  —  3  —  4  —  3  —  3—4? 

2. 

100  —  3  —  5  —  2  —  3  —  5  —  2  —  3? 

3. 

80  —  2  —  4  —  4  —  2  —  4  —  4  —  2?                          « 

4. 

40  —  8  —  2  —  5  —  5  —  2  —  4  —  3? 

5. 

70  —  6  —  4  —  2  —  8  —  3  —  5  —  2? 

6. 

100  —  3  —  4  —  3  —  10  —  5  —  5  —  3? 

7. 

25  +  3  +  2  +  8  —  5  —  3  —  6  —  4? 

8. 

83  +  7  +  2+8  —  3  —  7—5  —  5 

9. 

63  +  4  +  2+1  +  2  +  4  —  6  —  5? 

10. 

90  —  8  —  2—3  —  7—9+6+3? 

J.  1. 

One  day,  George  was  reckoning  up  his  money.    He  said  his 

father  gave  him  25  cents  at  one  time,  and  5  at  another ;  that  his  |  j 

mother 

gave  him  3  cents  at  one  time,  and  7  at  another ;  and  that 

his  uncle  Rufus  gave  him  6  cents  at  one  time,  and  3  cents  at  an-  1 1 

.    other. 

How  many  cents  had  he  ? 

2    Samuel  think3  that  the  sum  of23  +  4  +  3  +  G  +  4  +  4  +  5,  is   | 

LESSON     TWENTY-SECOND.  51 

49,  while  William  thinks  it  is  48,  and  Lydia  thinks  it  is  47.    How 
much  do  you  think  it  is  ? 

3.  Sarah  had  50  cents ;  she  spent  10  cents  for  ribbon,  4  cents 
for  sewing-silk,  and  5  cents  for  needles.  How  many  cents  had 
she  left  ? 

4.  As  Eliza  was  picking  up  shells  on  the  sea-shore,  she  found 
15  in  one  place,  5  in  another,  3  in  another,  7  in  another,  and 
enough  to  make  up  40  in  another.  How  many  did  she  find  in  the 
last  place  ? 

5.  Erastus  and  Edwin  played  *'  odd  or  even,"  beginning  their 
game  with  20  grains  of  corn  a-piece.  The  first  time  Erastus  won 
3  grains  from  Edwin,  the  second  time  he  won  3  grains,  and  the 
third  time  he  won  4.     How  many  had  each  boy  then  ? 

6.  A  party  of  hunters,  on  counting  their  game,  found  that  they 
had  shot  23  pigeons,  7  partridges,  20  quails,  3  woodcocks,  and  3 
snipes.     How  many  birds  had  they  shot  in  all  ? 

1:  Laura  found  44  blackberries  in  one  place,  6  in  another,  2  in 
another,  8  in  another,  and  9  in  another,  when,  feeling  tired,  she 
sat  down  to  rest.  She  ate  9,  and  put  4  in  a  hole  for  a  squirrel  to 
eat,  and  threw  C  to  some  birds.  She  then  found  20,  and  started 
for  home,  but  on  her  way  she  unfortunately  lost  8.  How  many 
had  she  to  carry  home  ? 

8.  Mr.  Day  went  out  to  pay  some  debts  that  he  owed,  and  to 
collect  some  money  that  was  due  him,  taking  with  him  30  dollars. 
He  paid  7  dollars  to  a  shoemaker,  3  dollars  to  a  laborer,  and  5 
dollars  to  a  hatter.  He  then  received  5  dollars  from  Mr.  Baker, 
30  dollars  from  Mr  Smith,  and  8  dollars  from  Mr.  Sumner,  after 
which  he  paid  Mr.  Gay  6  dollars  for  groceries,  and  2  dollars  for 
cloth,  and  Dr.  Fogg  7  dollars  for  services  as  a  physician.  How 
much  money  had  he  left  ? 


52 

C  0  L  B  U  R  X 

S    FIRST     PART. 

LESSO]^  XXIII. 

A.  1. 

9+8? 

6 

6+89! 

2. 

19  +  8? 

7. 

4+7? 

3. 

49+8? 

8. 

14+7? 

4. 

6+9? 

9 

4+37? 

6. 

36+9? 

10. 

4  +  67? 

B.  1. 

6  +  8+9? 

7. 

6  +  -x-  =  14? 

2. 

16  +  8  +  9? 

a 

6  +  *  =  34  ? 

3. 

66  +  8  +  9  ? 

9. 

46  +  *  =  54? 

4. 

9  +  7  +  8? 

10 

3  +  ^  =  11? 

5. 

89+7+8? 

11. 

13  +  ^  =  21? 

6. 

9+77  +  8? 

12. 

3+*==  61? 

C.  1. 

13  —  8? 

6. 

94  —  6? 

2. 

23  —  8? 

7. 

11—5? 

3. 

93  —  8? 

8. 

81—5? 

4. 

14  —  6? 

9. 

41  —  5? 

6. 

44  —  6? 

10. 

91  —  5? 

To  THE  Teacher.  —  Vary  and  extend  the  preceding  exercises  till 
the  scholars  appreciate  the  connexion  between  9  +  8, 15  +  8,  29  -t-  8,  <fcc., 
12  +  9,  23  +  9,  83  +  9,  &c.,  and  understand  fully  that  as  9^4-  8  =  17,  or 
7  more  than  10,  so  49  +  8  =  7  more  than  50,  or  57  ;  that  as  13  —  9  =  4, 
go  23  —  9  =  14,  83  —  9  =  74,  &c.    The  great  objects  to  be  aimed  at  are 
accuracy  and  promptness,  the  latter  being  scarcely  less  important  than 
the  former. 

r 

1 

LESSON    T  W  E  N  T  y  -  X  II I  11  D  . 

1' 

58 

D.  1.     48  +  4? 

12. 

73+9  +  6  +  9 

9 

2.     37  +  6? 

13. 

28+5+8+7 

+  4? 

3.     29  +  7? 

14. 

49+7  +  6+8 

? 

4.     53  +  8? 

15. 

67+8  +  4  +  9^ 

6.     74+7? 

16. 

67+8+4+6 

? 

6.     23+9+5  +  8? 

17. 

57  +  6—2  —  8: 

7.     2 

7  +  6  +  8+5? 

18. 

67+8  —  9  —  6? 

8.-  34+8+9—7? 

19. 

33+8+4+9+7+6+5+3  : 

9.     26+3+5—6? 

20. 

56+6+9+3+8+5+4+6  ? 

10.     2 

5+8+4-9  ? 

21. 

45+8+9+7+4+9+6+5  ? 

11.     29+7+9+4  ? 

22. 

38+6+7+7+5+8+9+8  ? 

E.  Find  the  sum  of  the  following  columns : —          j 

1 

2 

3 

4 

5 

6 

2 

9 

3 

3 

7 

7 

5 

6 

9 

9 

A 

9 

6 

6 

A 

B 

7 

A 

2 

3 

5 

6 

5 

7 

§ 

9 

9 

B 

5 

6 

3 

3 

7 

A 

2 

3 

9 

2 

A 

6 

6 

5 

3 

S 

A 

7 

9 

6 

3 

3 

9 

A 

7 

9 

— 

— 

— 

— 

— 

6* 


54  colburn's   first   part. 


To  THE  Teacher. — It  will  be  a  valuable  exercise  for  the  pupil  to 
count  by  twos,  threes,  fours,  &c.,  i.  e.,  to  call  the  results  obtained  by 
successive  additions  of  the  same  number  to  itself,  or  some  other  num- 
ber, till  all  possible  combinations  are  exhausted. 

Thus,  in  adding  threes,  we  shall  exhaust  the  varieties  of  combination 
by  beginning  thus  ;  three,  six,  nine,  twelve,  &c.  ;  two,  five,  eight, 
ELEVEN,  &c. ;  ONE,  FOUR,  SEVEN,  TEN,  <fcc.  A  similar  course  may  be 
taken  in  subtraction. 

F.  1.  Henry  bad  42  cents.  He  earned  9  cents  hy  doing  errands, 
5  cents  by  holding  a  gentleman's  horse,  and  8  cents  by  delivering 
a  letter,  after  which  he  spent  7  cents.   How  many  cents  had  he  left  ? 

2.  A  newsboy  bought  8  copies  of  the  Boston  Post,  9  of  the 
Atlas,  10  of  the  Traveller,  7  of  the  Bee,  10  of  the  Journal,  and  1 
of  the  Transcript.  How  many  papers  did  he  buy  in  all  ?  He  sold 
all  but  7  of  them.     How  many  did  he  sell  ? 

3.  A  trader  bought  8  yards  of  cloth,  for  which  he  paid  13  dol- 
lars ;  6  yards  for  which  he  paid  9  dollars,  9  yards  for  which  he 
paid  8  dollars,  and  6  yards  for  which  he  paid  8  dollars.  How 
many  yards  of  cloth  did  he  buy  in  all?  How  many  dollars  did 
he  pay  for  it  ? 

4.  A  man  bought  a  horse  for  G3  dollars,  and  was  obliged  to  sell 
him  for  8  dollars  less  than  he  cost  him.  For  how  much  did  he 
sell  him  ? 

5.  A  farmer  who  had  33  bushels  of  corn,  sold  6  bushels  for  5 
dollars.     IIow  many  bushels  had  he  left? 

6.  Mr.  Adams  owned  40  acres  of  land,  and  bought  enough  to 
make  up  54  acres.     How  many  did  he  buy  ? 

7.  Sarah  had  32  roses.  She  gave  9  to  one  of  her  companions, 
8  to  another,  and,  when  she  had  given  some  to  another,  she  had 
7  left.     How  many  did  she  give  to  the  last? 

8.  A  trader  bought  a  lot  of  grain  for  54  dollars,  and  it  cost  him 
3  dollars  more  to  have  it  carried  to  his  store.  For  how  much 
must  he  sell  it  to  gain  9  dollars  ? 


LESSON    TWENTY-FOURTH.  65 


9.  Mr.  Edwards  sold  a  colt  for  57  dollars,  a  sheep  for  8  dollars, 
a  calf  for  5  dollars,  and  a  cow  for  30  dollars,  and  in  part  payment 
received  a  horse  worth  93  dollars.    How  much  still  remained  due  ? 

10.  Mr.  Boy  den  and  Mr.  Manchester  each  bought  a  yoke  of 
oxen.  Mr.  Boyden  gave  in  paj^ment  for  his  oxen,  a  cart  worth 
47  dollars,  1  ten-dollar  bill,  1  five-dollar  bill,  1  three-dollar  bill,  7 
one-dollar  bills,  and  9  silver  dollars.  Mr.  Manchester  gave  in 
payment  for  his  oxen,  a  cow  worth  30  dollars,  a  double  eagle 
worth  20  dollars,  an  eagle  worth  10  dollars,  a  half-eagle  worth  5 
dollars,  an  ox-yoke  worth  10  dollars,  and  9  dollars  worth  of  hay. 
Which  paid  the  most  for  his  oxen,  and  how  much  the  most  ? 


LESSON  XXIV. 

A.  1.     30+50?  7.     40  +  20-1-20? 

2.  32  +  50?  8.     43  +  20  +  20? 

3.  39  +  50?  9,     40  +  28  +  20? 

4.  20  -^  60?  10.     30  +  20  +  40 ? 

5.  25  +  60?  11.     38  +  20  +  40? 

6.  20  +  68?  12.     30+20  +  47? 

B.  1.  How  many  are  24  -f  67  ? 

SoLUTioJf.--24  and  60  are  84,  and  7  are  91. 

2.  63  +  29?  11.     27  +  58+12? 

3.  26+55?  12.     33  +  47  +  16? 

4.  37+48?  13.     24  +  29+47? 

5.  24  +  37?  14.     24  +  29+37? 

6.  73  +  19?  15.     27  +  27+27? 

7.  28+53?  16.     11  +  46  +  25? 

8.  37  +  67?  17.     34  +  26  +  27? 

9.  29  +  29?  18.     16  +  17+19? 


56  colburn's  first   part. 


-\] 


C.  1.  80—20?  7.  90  —  80? 

2.  86  —  20?  8.  97  —  80? 

3.  60  —  30?  9.  80  —  40? 

4.  67  —  30?  10.  86  —  40? 

5.  70  —  40?  11.  60  —  30? 

6.  77  —  40?  12.  53  —  30 

D.  1.  68-26? 

Solution. — 68  minus  20  are  48,  minus  6  are  42. 

2.  43  —  17?  7.     81  —  23? 

3.  92  —  67?  8.     52  —  27? 

4.  83  —  48?  9.     48  —  29? 

5.  61  —  23?  10.     97  —  58? 

6.  56  —  19? 

E.  1.     63  +  37  —  82?  4.     64  +  36  —  48? 

2.  48  +  35  —  27?  5.     25  +  39—42? 

3.  24  +  67  —  19?  6.     27  +  64  —  18? 

F.  1.  Joseph  bought  a  "First  Book  of  Arithmetic"  for  25 
cents,  and  a  slate  for  13  cents.     How  much  did  he  pay  for  both  ? 

2.  Martha's  mother  gave  her  75  cents  with  which  to  purchase 
school-books  and  paper.  She  bought  a  Primary  Geography  for 
37  cents,  a  Spelling  Book  for  17  cents,  and  spent  the  rest  of  her 
money  for  paper.     How  much  did  she  spend  for  paper  ? 

3.  A  farmer  sold  a  horse  for  93  dollars,  which  was  26  dollars 
more  than  he  gave  for  him.     How  much  did  he  give  for  him  ? 

4.  A  horse  dealer  bought  a  horse  for  54  dollars,  and  after  pay- 
ing 17  dollars  for  keeping  him,  he  sold  him  for  96  dollars.  How 
much  did  he  gain  by  the  transaction  ? 


LESSON    TWENTY-FOURTH.  57 


5.  A  man  bought  a  sleigh  for  21  dollars.  He  paid  9  dollars  for 
painting  and  repairing  it,  and  then  gave  it  and  18  dollars  in  mo- 
ney for  another  sleigh.  How  much  did  the  second  sleigh  cost 
him? 

6.  From  a  cask  containing  64  gallons  of  oil,  18  gallons  were 
drawn  out  at  one  time,  and  25  at  another,  after  which  17  gallons 
were  put  in.     How  many  gallons  were  then  in  the  cask  ? 

7.  There  were  18  sheep  in  one  flock,  27  in  another,  and  39  in 
another ;  but  at  night  they  were  all  put  into  the  fold.  How  many 
were  there  in  the  fold  ?  The  next  day,  23  were  driven  to  one 
pasture,  26  to  another,  and  the  rest  to  another.  How  many  were 
driven  to  the  last  pasture  ? 

8.  Ralph  shot  27  pigeons,  15  partridges,  14  woodcocks,  and  as 
many  quails  as  there  were  partridges  and  woodcocks  together. 
How  many  quails  did  he  shoot?     How  many  birds  in  all? 

9.  Mr.  Thompson  owes  13  dollars  to  Mr.  Baker,  9  dollars  more 
to  Mr.  Ellis  than  to  Mr.  Baker,  and  as  much  to  Mr.  French  as  he 
owes  to  Mr.  Thompson  and  Mr.  Ellis  together.  How  much  does 
he  owe  to  each,  and  how  much  to  all  ? 

10.  Mr.  Talbot  bought  a  large  lot  of  apples.  His  son  George  asked 
how  much  they  cost  him,  to  which  he  replied:  "  I  paid  17  dollars 
in  silver,  25  in  gold,  and  13  dollars  more  in  bank  bills  than  in 
silver  and  gold  together.  Now,  if  yon  will  tell  me  what  they  cost, 
I  will  give  you  the  difference  between  their  cost  and  100  dollars." 
George  answered  correctly.  What  was  his  answer,  and  how  much 
money  did  his  father  give  him  ? 


58  colbukn's  first   part. 


LESSON  XXV. 

A.  The  numbers  above  one  hundred  are  counted 
thus :  —   ' 

One  hundred  one,  one  hundred  two,  one  hundred  three,  Jfc,  to  one 
hundred  ninety-eight,  one  hundred  ninety-nine,  two  hundred,  two  hun* 
dred  one,  two  hundred  two,  two  hundred  three,  ^'C,  to  ten  hundred, 
which  is  generally  called  one  thousand,  ten  hundred  one,  or  one 
thousand  one,  ten  hundred  ttvo,  or  one  thousand  two,  &c.,  to  elei^en 
hundred  one,  or  one  thousand  one  hundred  one,  &c.,  to  nineteen  hun- 
dred ninety-nine,  or  one  thousand  nine  hundred  ninety-nine,  twenty 
hundred,  or  two  tltousand. 

Ten  hundred,  or  one  thousand  is  written  1000.  Twenty  hun- 
dred, or  two  thousand,  is  written  2000.  Thirty  hundred,  or  three 
thousand,  is  written  3000.  Ninety  hundred,  or  nine  thousand,  is 
written  9000. 

The  following  exercises  suggest  the  manner  of  reading  and 
writing  numbers  above  one  hundred :  — 

100  -f     2  =  102.  1000  -f    10  ==  1010. 

400  -f     9  =  409.  1100  -f.      3  =  1103. 

100  -f  10  =  110  1000  4-    28  =  1028. 

100  +  11  =  111.  1100  4-    11  =  1111. 

300  4-12  =  312.  11004-    17=1117. 

600  4-  20  =  620.  1000  4-  117  =  1117. 

100  4-  29  =  129.  3200  4-    20  =  3220. 

1000  4-     1  =  1001.  4200  4-    34  =  4234. 

1000  4-     4  ==  1004.  4000  4-  234  =  4234. 


LESSON    TWENTY-SIXTH.  69 


B.  Eead  the  following  numbers :  — 

/.     A27  5.     5^ A  p.     //i'cf 

2.    ^6B        6.  s^oy      ^0     6oo6 

3.     S60  7.    ^5S6        //.     S22A 

A.    630         8.    37M       ^^'    ^^'^7 

C.  Write  each  of  the  following  in  figures  :  — 

1.  Three  hundred  twenty-seven. 

2.  Eight  hundred  four. 

3.  Seventeen  hundred  twenty-eight. 

4.  Forty-six  hundred  thirty-six. 

5.  Four  thousand  six  hundred  thirty-six. 

6.  Twenty-six  hundred  six. 

To  THE  Teacher.  —  For  other  exercises  in  Notation  and  Numera- 
tion, see  Arithmetic  and  its  Applications. 


LESSON   XXVI. 

Addition. 

A-  All  such  questions  as  "  How  many  are  6  -|-  ^  +  ^  ?" 
<<4^8-|-9?"  &c.,  are  questions  in  Addition.  We  are  required 
in  the  first  to  add  6,  9,  and  7  together ;  and  in  the  second,  to  add 
4,  8,  and  9.  It  is  obvious  that  in  each,  we  are  required  to  find  a 
number  equal  in  value  to  all  the  given  numbers.  Thus,  in  the 
first  question,  we  are  required  to  find  a  number  equal  in  value  to 
6-f  9-f  7. 

Addition  is  a  process  by  which  we  find  a  number  equal  in  value  to 
several  given  numbers. 


60  colburn's   first   part. 

The  number  tlms  found  is  called  the  sum  or  amount  of  the 
I  given    numbers.     Thus    the    sum    of    C,    9,    and   7   is   22,    for 
6-1-9  +  7=- 22. 

B.  When  writing  large  numbers  for  addition,  we  place  them  in 
a  column,  so  that  the  figures  of  the  same  denomination  shall  come 
under  each  other,  i.  e.,  so  that  units  shall  come  under  units,  tens 
under  tens,  &c.  We  then  begin  at  the  right  hand,  and  add  the 
columns  separately,  as  in  the  following  examples  :  — 

1.  What  is  the  sum  of  723  +  896  +  589  -j-  967  ? 

Solution.  —  Writing  the  numbers  as  opposite,  we  first  '/^  ^  9 

add  the  units'  column,  7  +  9-f-6-J-3==25  units  ==  2  tens  ' 

and  5  units.  Writing  5  units,  and  adding  2  tens  to  the 
tens'  column,  we  have  2-|-6-|-8  +  9-|-2  =  27  tens  =  2 
hundreds,  and  7  tens.  Writing  7  tens,  and  adding  2  hun- 
dreds to  the  hundreds'  column,  we  have  2  -f-  9  -j-  5  -f" 
8  4-7  =  31  hundreds,  which,  being  the  sum  of  the  last 
column,  we  write.     The  answer,  therefore,  is  3175. 

3/75 

To  test  the  correctness  of  the  work,  examine  it  carefully  to  see 
if  any  error  can  be  detected.  Or,  add  the  numbers  again,  begin- 
ning at  the  top  of  the  column. 

To  THE  Teacheu. — The  design  of  thi^  work  renders  it  impracticable 
to  give  further  illustrations  here,  but  the  Teacher  can  readily  supply 
them  if  they  are  needed  by  the  class.  (See  Arithmetic  and  its 
Applications,  Sect.  IV.) 

C.  Add  the  following :  — 

12  3  4  6 

SAP  A/7  SS7  Sp6  BA3 

37  s  2S6  6/3  /7B  2AB 

6 AS  A3p  ASB  S5p  537 

376  67P  85  A  A3B  AS>p 


LESSON    TWENTY-SEVENTH.  61 


6.  4254-487  +  569  +  837+694? 

7.  854  +  308  +  560  +  716  +  593  ? 

8.  672  +  481  +  326  +  425  +  519  ? 

9.  243  +  495  +  826  +  324  +  476  ? 
10.  627+  756  +  434+  874+999  ? 


LESSON  XXVII. 

Subtraction. 

A.  Such  questions  as  "  4  from  9?"  "  12  —  6?"  "How  many 
more  are  17  than  8?"  &c.,  are  questions  in  Subtraction. 

"We  are  required,  in  the  first,  to  subtract  4  from  9 ;  in  the 
second,  to  subtract  6  from  12;  and  in  the  third,  to  subtract  8 
from  17. 

Subtraction  is  a  process  by  which  we  find  the  difference  between  two 
numbers,  or  the  excess  of  one  number  over  another. 

The  larger  of  the  two  given  numbers  is  called  the  minuend,  the 
smaller,  or  one  to  be  subtracted,  is  called  the  subtrahend,  and 
the  answer  is  called  the  difference,  or  remainder. 

B.  We  write  large  numbers  for  Subtraction,  so  that  figures  of 
the  same  denomination  shall  come  under  each  other,  and  subtract 
as  illustrated  in  the  following  examples :  — 

1.  How  much  is  8436  —  6122? 

Solution. — Writing  the  numbers  as  opposite,  we  have  0  /  0/j 
2  units  from  6  units  leave  4  units;  2  tens  from  3  tens 

leave  1  ten ;  1  hundred  from  4  hundreds  leaves  3  hun-  S / Q9 
dreds  ;  6  thousands  from  8  thousands  leaves  2  thousands. 


Therefore,  the  answer  is  2  thousands,  3  hundreds,  1  ten, 

and  4  units,  or  2314.  23 if  A 


62  coLB urn's   first  part. 


Ill  the  same  manner  perform  tlie  following  :  — 

2.  4893  —  1231?  5.     4867—1614? 

3.  5987  —  3125?  6.     9318  —  2106? 
4     8958  —  6713?  7.     6985  —  1401? 

C.  If  a  figure  of  the  subtrahend  is  larger  than  the  corresponi- 
ing  figure  of  the  minuend,  we  take  one  of  the  next  higher 
denomination  of  the  minuend,  and  reduce  (t.  e..  change)  it  to  the 
required  denomination,  as  in  the  following  example  : — 

1.  How  much  is  947  —  458  ? 

Solution. — As  we  cannot  subtract  8  units  from  7  units,  wo.  take  one 
of  the  4  tens  (leaving  3  tens),  and  reduce 

0    V  o  y/'V  _  f  minuend  changed  in  form  to   show 
OVOj//       I     the  reduction. 

Q     j^      y^  *="   minuend. 
ji-     S      O  ^   subtrahend. 


^     O      p  =•  quotient. 


to  its  value  in  units :  1  ten  =  10  units,  which,  added  to  the  7  units, 
gives  17  units  :  17  units     -  8  units  =  9  units. 

As  we  cannot  take  5  tens  from  3  tens,  we  take  one  of  the  9  hundreds, 
leaving  8  hundreds,  and  reduce  it  to  its  value  in  tens  :  1  hundred  =  10 
tens,  which,  added  to  the  3  tens  =  13  tens  ;  13  tens  —  5  t-ens  =  8  tens. 

4  hundreds  from  8  hundreds  leave  4  hundreds.  Therefore,  947  — 
458  =  4  hundreds,  8  tens,  and  9  units,  or  489. 

To  prove  the  correctness  of  the  answer,  add  the  subtrahend 
and  remainder  together;  if  their  sum  is  equal  to  the  minuend, 
the  work  is  correct ;  if  not,  there  is  an  error  in  the  subtraction  or 
the  addition,  and  the  work  should  be  re-examined  to  detect  it. 

To  THE  Teacher. — For  more  full  illustrations,  see  Arithmetic  and 
ITS  Applications. 


LESSON    TWENTY-EIGHTH.  63 


2.  48G4  — 2579?  8.  5426  —  3987? 

3.  8149  — 34G3?  9.  9943  —  4399? 

4.  2769  —  1487?  10.  9333  —  8888? 

5.  2144  —  1397?  11.  4634  —  2359? 

6.  8432  —  3586?  12.  9257  —  4328? 

7.  4374  —  5856?  13.  8642  —  5853? 


LESSON  XXVIII. 

The  method  of  writing  numbers  by  figures  is  called  the  Arabic 
Method. 

There  is  a  method  of  expressing  numbers  by  letters,  called  the 
Roman  Method.  The  letter  I  stands  for  one,  V  for  five,  X  for  ten, 
L  for  fifty,  C  for  one  hundred,  D  for  five  hundred,  and  M  for  one 
thousand. 

If  a  letter  is  repeated,  it  indicates  that  the  number  for  which  it 
stands  is  repeated.  Thus :  I  stands  for  one,  II  for  two,  III  for 
three,  X  for  ten,  XX  for  twenty,  XXX  for  thirty,  CC  for  two 
hundred,  &c.,  &c. 

If  a  letter  representing  one  number  stands  before  a  letter, 
representing  a  larger  number,  the  value  of  the  formei  is  sub- 
tracted from  the  value  of  the  latter.  Thus :  IV  =  1  from  5  =  4, 
IX  =  1  from  10  =  9,  XL  =  10  from  50  =  40,  XC  =  10  from 
100  =  90,  &c. 

If  a  letter  representing  one  number  stand  before  a  letter  repre- 
senting a  smaller  number,  the  value  of  the  former  is  to  be  added  to 
the  value  of  the  latter.  Thus:  VI  =  5+  1  =  6,  XI  =  lO-f- 1  =  11, 
XV  =  10  -I-  5  =  15,  &c.     CX  =  100  -f  10  =  110.     Hence  — 


64  COLBURN'S    i'lRST     PART. 


1=1  XI  ==  11  XXI  =  21 

II  =  2  XII  =  12  XXIV  =  24 

III  =  3  XIII  =  13  XXV  =  25 

IV  =  4  XIV  ==  14  XXX  =  30 
V  =  5  XV  =  15  XXXIX  =  39 

VI  ==  6  XVI  =  16  XLIV  =  44 

VII  =  7  XVII  =  17  LXX  =  70 

VIII  =  8  XVIII  =  18  LXXXIX  =  89 

IX  ==  9  XIX  =  19  XC  =  90 

X  =  10  XX  =  20  CXXXIX  =  139 


LESSON  XXIX. 
Tables  of  Moneys,  Weights,  and  Measures. 

A.  The  money  we  use  is  called  United   States  or  Federal 
Money. 

TABLE   OF   UNITED    STATES   MONEY. 

10  mills  =  1  cent. 

10  cents  =  1  dime. 

10  dimes  =  1  dollar. 

10  dollars  =  1  eagle. 
The  coins  of  the  United  States  are :  the  cent,  the  three-cent 
piece,  the  half-dime,  worth  5  cents;  the  dime,  worth  10  cents; 
the  quarter-dollar,  worth  25  cents;  the  half-dollar,  worth  50 
cents ;  the  dollar,  worth  100  cents ;  the  three-dollar  piece ;  the 
eagle,  worth  10  dollars  ;  the  double-eagle,  worth  twenty  dollars  ; 
the  half-eagle,  worth  five  dollars ;  the  quarter-eagle,  worth  two 
and  a  half  dollars,  and  the  fifty-dollar  piece. 


*  In  reciting  these  tables,  let  the  pupils  say  "  equal"  in  place  of 
'*  make  one,"  the  phrase  often  used. 


LESSON    TWENTY-NINTH.  65 


The  character  $  placed  at  the  left  of  figures,  shows  that  they 
represent  dollars,  or  values  in  United  States  Money.  The  dollars 
are  alwa3*s  placed  at  the  left  of  the  decimal  point  (See  Lesson 
XXV),  and  the  cents  and  mills  at  the  right. 

Thus,  to  express  14  dollars  38  cents,  we  should  write  14  at  the 
left  of  the  decimal  point,  and  38  at  the  right  of  it.   Thus :  $14.38. 

Illustration. — $  8.27  =  8  dollars,  27  cents. 
$15.06  =  15  dollars,  06  cents. 
$2,327  •=»  2  dollars,  32  cents,  7  mills. 

Read  the  following :  — 

1.  $4.28.  3.     $82.36.  5.     $40.03. 

2.  $5.37.  4.     $75.07.  6.     $28.79. 

B.  The  money  used  in  England  is  called  English  or  Sterling 
Money. 

TABLE   OF   STERLING   MONEY. 

PULL  TABLE.  ABBREVIATED  TABLE. 

4  farthings  =  1  penny.  4  gr.  =  Id. 

12  pence  =  1  shilling.  12  d.  =  Is. 

20  shillings  =  1  pound.  20  s.  =  1£. 

The  English  pound  is  worth  about  $4.84. 

The  English  shilling  is  worth  about  a  quarter  of  a  dollar,  and  the 
English  penny  is  worth  about  2  cents.  The  term  shilling  is  sometimes 
used  in  New  York,  New  England,  and  some  other  States  of  the  Union, 
but  it  does  not  mean  an  English  shilling.  A  New  York  shilling  is 
worth  just  12J  cents.  A  New  England  shilling  is  worth  just  16§ 
cents.  The  ninepence  of  New  England  is  the  same  as  the  shilling  of 
New  York. 

C.  Iron,  flour,  sugar,  wool,  coal,  and  almost  all  articles  except 
gold,  silver,  and  jewels,  are  weighed  by  Avoirdupois  Weight. 

"^    6*  E 


66  colburn's  first  part. 

FULL  TABLE.  ABBREVIATED  TABLE. 

16  drams  =  1  ounce.  16  dr.  =  1  oz. 

16  ounces  =  1  pound.  16  oz.  =  1  lb. 

25  pounds  =  1  quarter.  25  lbs.  =  1  qr. 

4  quarters  =  1  hundred  weight.  4  qrs.  =  1  cwt. 

20  hundred  weight  =  1  ton.  20  cwt.=  1  T. 

N3TE. — Formerly  the  quarter  was  reckoned  at  28  lbs.,  the  hundred- 
weight at  112  pounds,  and  the  ton  at  2240  lbs.,  and  they  are  so  reck- 
oned at  the  present  time  in  Great  Britain,  and  at  the  United  States 
Custom  Houses.   Merchants  usually  reckon  them  as  given  in  the  table. 

D.  Gold,  Silver,  and  precious  stones  are  weighed  by  Troy 
Weight. 

FULL  TABLE.  ABBREVIATED  TABLE. 

24  grains  =  1  pennyweight.  24  gr.  =  1  dwt. 

20  pennyweights  =  1  ounce.  20  dwt.  =  1  oz. 

12  ounces  =  1  pound.  12  oz.  =  1  lb. 

E.  Apothecaries'  Weight  is  used  in  compounding  or  mixing 
medicines,  but  they  are  sold  by  Avoirdupois  weight. 

FULL  TABLE.  ABBREVIATED  TABLE. 

20  grains  =  1  scruple.  20  gr.  =19. 

3  scruples  =  1  dram.  3  9  =  1  5. 

8  drams  =  1  ounce.  8  .:^  =  1  5. 

12  ounces  =  1  pound.  12  g  ==  1  ib. 

F.     COMPARISON  OF  AVOIRDUPOIS,  TROY,  AND  APOTHE- 
CARIES' WEIGHT. 
A  pound  Avoirdupois  is  heavier  than  a  pound  Troy,  but  an 
ounce  Avoirdupois  is  not  so  heavy  as  an  ounce  Troy. 

Their  relative  weights  may  be  seen  in  the  following  table  of 
comparison,  which  expresses  the  value  of  each  in  grains  Troy ;  — 
1  lb.  Avoirdupois  =  7000  grains  Troy, 
lib.  Troy  =  1  lb.  =  5760       " 
1  oz.  Avoirdupois  =    437  J     ** 
1  oz.  Troy  =  Ig    =    480       " 
1  dr.  Avoirdupois  =      27J^^  " 
1  3  Troy  r=      60       " 

19"  =      20       *« 

Idwt.  =      24       " 

1  gr.  Apothecaries  =        1       ** 


LESSON    TWENTY-NINTH.  67 

It  follows,  then,  that  — 

144  lbs.  Avoirdupois  =  175  Troy. 
192  oz.  "  =  175  oz.  Troy. 

1  lb.  "  =  J-Jf  of  1  lb.  Troy. 

1  oz.  "  =  {.Jl  of  1  oz.  Troy. 

G.  Long  Measure  is  used  for  measuring  lengths  and  dis- 
tances. 

FULL  TABLE.  ABBREVIATED  TABLE. 

12  lines  =  1  inch.  12  1.  =1  in. 

12  inches  =  1  foot.  12  iR.=  1  ft. 

3  feet  =  1  yard.  3  ft.  =  1  yd. 

5 J  yards,  or  •.  5  J  yds.,  or  ^ 

y  =1  rod,  or  pole.  V  =  1  rd.  or  p. 

lejfeet  /  ^  16Jft.  /  ^ 

40  rods  =  1  furlong.  40  rds.  =  1  fur. 

8  furlongs  =  1  mile.  8  fur.  =  1  m. 

3  miles  =  1  league.  3  m.    =1  le. 

H.  Cloth  Measure  is  used  for  measuring  cloths,  silks,  &c. 

PULL  TABLE.  ABBREVIATED  TABLE. 

2J  inches  =  1  nail.  2J  in.  =  1  na. 

4    nails    =  1  quarter.  4   na.  =  1  qr. 

4    quarters  =  1  yard.  4   qr.  =  1  yd. 

I.  Square  Measure. — This  measure  is  used  in  measuring 
land,  and  all  kinds  of  surfaces. 

Preliminary  Defitiitions. — An  anc/le  is  the  diflference  in  direction 
of  two  lines.  The  point  where  the  lines  meet  is  called  tha  vertex 
of  the  angle. 

When  the  two  angles  formed  by  one  straight  line  meeting 
another  are  equal  to  each  other,  they  are  called  rfght  ai^gles. 


68  colburn's  first   part. 


One  line  i^  perpendicular  to  another  when  it  makes 'right  angles 
with  it. 

^-  The  angle  A  C  B  is  equal  to  the  angle 

BCD,  and  hence  they  are  right  angles. 
D.  Therefore,  B  C  is  perpendicular  to  A  D. 


An  angle  greater  than  a  right  angle,  is 
called  an  obtuse  angle,  and  an  angle  less  than  a  right  angle  is 
called  an  acute  angle. 

A  RIGHT  ANGLE.  AN  ACUTE  ANGLE.  AN  OBTUSE  ANGLE. 


A  four-sided  figure  having  all  of  its  angles  right  angles,  is  called 
a  rectangle. 

A  rectangle  having  all  of  its  sides  equal,  is  called  a  square.  A 
square,  then,  has  four  equal  sides,  and  four  equal  angles. 

A  square  foot  is  a  square  measuring  one  foot  on  everj  side.  A 
square  yard  is  a  square  measuring  a  yard  on  every  side,  &c. 

TABLE  OF  SQUARE  MEASURE. 

FULL  TABLE.  ABBREVIATED  TABLE. 

144  square  inches  =  1  square  foot.     144  sq.  in.  =  1  sq.  ft. 

9  square  feet  =  1  square  yard.  9  sq.  ft.  =  1  sq.  yd. 

30J-  square  yards,  or  >»  30J  sq.  yds.  or  >| 

}.  =  1  sq.  rod.  V  =1  sq.  rd. 

272J  square  feet,         i  272J  sq.  ft.      J 

40  square  rods  =  1  rood.  40  sq.  rds.  =  1  R. 

4  roods  =  1  acre.  4  R.  =  1  A. 

040  acres  =  1  square  mile.  640  A.  =  1  sq.  m. 

J.  Cubic  Measure. — Cubic  Measure  is  used  in  measuring 
solids. 

A  solid  is  a  magnitude  which  has  length,  breadth,  and  thickness. 


LESSON    TWENTY-NINTH.  69 


A  cube   is  a  rectangular  solid,  whose   length,    breadth,  and 

height,  are  equal.  It  may  also  be  defined  as  a  solid  bounded  by 
six  equal  squares. 

A  cube  1  foot  long,  1  foot  wide,  1  foot  high,  would  be  a  cubic 
foot. 

A  cube  1  yard  long,  1  yard  high,  and  1  yard  wide,  would  be  a 
cubic  yard. 

FULL  TABLE.  ABBREVIATED  TABLE. 

1728  cubic  inches  =  1  cubic  /foot.  1728  cu.  in.  =  1  cu.  ft. 

27  cubic  feet  =  1  cubic  yard.  ^^  ^-  ^^-  ^^  ^  ^^-  y^' 

16  cubic  feet  =  1  cord  foot.  IG  cu.  ft.  =  1  cd.  ft. 

8  cord  feet,  or  ^  8  cd.  ft.  or 
I  =1  cord  wd. 

128  cubic  feet       J  128  cu, 


W      V.^^.      il.     Ul      -\ 

1  cord  wd.  y  =1  cd.  wd. 

.  ft.     / 


K.  Circular  or  Angular  Measure.  —  Circular  or  Angular 
Measure  is  used  to  measure  angles,  and  the  circumferences  of 
circles. 

A  circle  is  a  surface  bounded  by  a  curved  line,  which  is  every- 
where equally  distant  from  a  point  within,  called  the  centre.  The 
boundary  line  is  called  the  circuTfiference  of  the  circle. 

The  figure  represents  a  circle,  of  which  C. 
is  the  centre. 

The  distance  from  the  centre  of  a  circle  to 
the  circumference  is  called  the  radius. 

The  distance  from  a  point  on  one  side  of  a 
circle  through  the  centre  to  a  point  on  the 
opposite  side  is  called  the  diameter.     Any  portion  of  the  circum- 
ference is  called  an  arc. 

Every  circumference  of  a  circle,  whether  large  or  small,  is  sup- 
posed to  contain  360  equal  parts,  called  degrees.  Each  degree  is 
divided  into  60  equal  parts,  called  minutes,  and  each  minute  into 
60  equal  parts,  called  seconds. 


70  colburn's   first   part. 


A  degree  may  be  considered  simply  as  the  360th  part  of  the 
circumference  of  thz  circle  considered.  Hence  its  length,  as  well 
as  that  of  its  subdivisions,  must  vary  with  the  size  of  the  circle. 

FULL  TABLE.  ABBREVIATED  TABLE. 

60  seconds  =  1  minute.  60''''  =  V. 

60  minutes  =  1  degree.  60-^  =  1°. 

360  degrees  =  1  circumference.  860°  =  1  circ. 

L.     Dry  Measure  is  used  for  measuring  grain,  nuts,  salt,  &c. 

PULL  TABLE.  ABBREVIATED  TABLE* 

2  pints  =  1  quai|p  2  pts.  =  1  qt. 

8  quarts  =  1  peck.  8  qts.  =  1  pk. 

4  pecks  =  1  bushel  4  pks.  =  1  bu. 

The  chaldron  of  36  bushels  is  sometimes  used  in  measuring  coal. 
Ch.  is  the  sign  for  chaldron. 

The  bushel  contains  2150|  cubic  inches,  and  the  quart  contains 
67^  cubic  inches. 

M.     All  kinds  of  liquids  are  measured  by  Liquid  Measure. 

FULL  TABLE.  ABBREVIATED  TABLE. 

4  gills  =  1  pint.  4  gls.  =  1  pt. 

2  pints  =  1  quart.  2  pts.  =  1  qt. 

4  quarts  =  1  gallon.  4  qts.  =  1  gal. 

The  hogshead  of  68  gallons  is  used  in  estimating  the  contents 
of  reservoirs,  or  other  large  bodies  of  water ;  but  in  all  other  cases 
the  term  hogshead  is  not  a  definite  measure.     Casks  containing 
from  50  or  60,  to  100  or  200  gallons,  are  called  hogsheads. 
A  barrel  of  cider  is  usually  reckoned  at  81 J  gallons. 
The  gallon  contains  231  cubic  inches. 

The  beer  gallon  is  sometimes  used  in  measuring  beer,  milk,  and 
ale.  It  contains  282  cubic  inches,  and  the  beer  quart  contains 
70J  cubic  inches. 


LESSON    TWENTY-NINTH.  71 


N.     COMPARISON  OF  DRY,  LIQUID,  AND  BEER  MEASURE. 
1  qt.  dry  measure  =  67^  cubic  inches. 
1  qt.  liquid  measure  =r  57J  cubic  inches. 
1  qt.  beer  measure  ==  70J  cubic  inches. 

0.      TABLE   OF   TIME. 

FULL  TABLE.  ABBREVIATED  TABLE. 

60  seconds  =  1  minute.  60  sec.  =  1  min. 

60  minutes  =  1  hour.  60  min.  =  1  h. 

24  hours  =  1  day.  24  h.  =  1  d. 

7  days  =  1  week,  7  d.  =  1  wk. 

865  days,  or  52  >.  3G5  d.  or  52  wk, 


>l  ooo  a.  or  i)^  WK.  % 

1=1  year.  ^^^  |  =  1  y. 


weeks,  IJ  days.   ^  l^  d. 

To  avoid  the  inconvenience  of  reckoning  J  day  with  each  year, 
every  fourth  year  (called  leap  year)  is  reckoned  at  366  days,  and 
the  others  at  365. 

The  year  is  divided  into  12  months,  which  differ  somewhat  in 
length,  as  is  seen  in  the  following 

TABLE     OF    MONTHS. 

January  has  31  days.  July  has  31  days. 

February  has  28  days.*  August  has  31  days. 

March  has  31  days.  September  has  30  days. 

April  has  30  days.  October  has  31  days. 

May  has  31  days.  November  has  30  days. 

June  has  30  days.  December  has  31  days. 

*  Except  in  leap  year,  when  it  has  29. 


72  colburn's   first  part. 

p.      MISCELLANEOUS. 

12  things  =  1  dozen. 

12  dozen  =  1  gross. 

12  gross   =  1  great  gross, 

20  tilings  =  1  score. 
A  barrel  of  beef  or  pork  "weighs  200  lbs. 
A  barrel  of  flour  weighs  196  lbs. 

,        PAPER. 

24  sheets  =  1  quire. 

20  quires  =  1  ream. 

BOOKS. 

A  sheet  folded  in  2  leaves  is  called  a  folio. 

"     ♦*         «•       "   4       «*     **         "  quarto,  or  4to. 

"     "        ««      «*   8      "     "        "  octavo,  or  8vo. 

"     «*        "      "  12      "     "        "  duodecimo  or  12mo. 

"     "        "      *<  18      "     "        "  18mo. 
This  book  is  a  duodecimo. 

Q.      FRENCH    MEASURES    AND    WEIGHTS. 

The  folio-wing  measures  and  weights  are  often  referred  to  in 
this  country,  especially  in  scientific  works  : 

rRENCH   LONG   MEASURE. 

10  millimetres  =  1  centimetre. 
10  centimetres  =  1  decimetre. 
10  decimetres  =  1  metre. 
10  metres         =  1  decametre. 


LESSON    TWENTY-NINTH.  73 


10  decametres  =  1  hectometre. 

10  hectometres  =  1  kilometre. 

10  kilometres    =  1  myriametre. 
The  metre  is  regarded  as  the  unit  of  measure,  and  equals  39.371 
of  our  inches.     It  is  the  twenty-millionth  part  of  the  distance 
measured  on  the  meridian,  from  one  pole  to  the  other. 

FRENCH    WEIGHTS. 

10  milligrammes    =  1  centigramme. 
10  centigrammes  ==  1  decigramme. 
10  decigrammes    =  1  gramme. 
10  grammes  =  1  decagramme. 

10  decagrammes   =  1  hectogramme. 
10  hectogrammes=  1  kilogramme. 
10  kilogrammes    =  1  myriagramme. 

The  gramme  is  regarded  as  the  unit  of  this  weight,  and  equals 
about  IS.yyy*^  grains  Troy. 

The  kilogramme  is  the  weight  most  frequently  used  in  business 
transactions,  and  equals  very  nearly  2^  pounds  Avoirdupois. 

FRENCH  MONEY. 

10  centimes  =  1  decime. 

10  decimes  =  1  franc. 

The  franc  equals  18|  cents,  and  the  five-franc  piece  often  seen 
in  the  United  States,  is  equal  m  value  to  93  cents. 


74 

colburn's 

FIRST    PART. 

LESSON  XXX. 

A.      TABLE 

• 

Add  10  twos  together. 

2  times  1,  or  once  2=2 

2  times  6,  or  6  times  2  = 

12 

2  times  2                       =4 

2  times  7,  or  7  times  2  = 

14 

2  times  3,  or  3  times  2=6 

2  times  8,  or  8  times  2  = 

16 

2  times  4,  or  4  times  2  =    8 

2  times  9,  or  9  times  2  = 

18 

2  times  5,  or  5  times  2  =  10 

2  times  10,  or  10  times  2= 

.20. 

B.  1. 

*  times  7  =  14? 

6. 

4  =  *  times  2  ? 

2. 

*  times  2  =  14? 

7. 

20  =  *  times  2  ? 

3. 

•je  times  4  =    8  ? 

8. 

16  =  -jv  times  2  ? 

4. 

*  times  2  =  12  ? 

9. 

10  =  •}«•  times  5  ? 

5. 

*  times  2  =  18? 

10. 

6  =  *  times  2  ? 

C.  1. 

4  times  ^  =  8? 

6. 

10  =  2  times  *  ? 

2. 

2  times  «  =  8  ? 

7. 

4  =  2  times  *  ? 

3. 

2  times*  =  10? 

8. 

20  =  2  times  *  ? 

4. 

2  times  *  =  6  ? 

9. 

12  =  2  times  *  ? 

6. 

2  times  *  =  14  ? 

10. 

18  =  2  times  *  ? 

D.  1. 

8  times  2,  plus  4  =  ^f 

times  10? 

2. 

2  times  5,  plus  8  =  * 

times  2 

,? 

3. 

7  times  2,  minus  6  = 

*  times  4? 

1 

4. 

2  times  9,  minus  6  = 

*  times  2  ? 

5. 

2  times  4,  plus  8  =  * 

times  2  ? 

LESSON    THIRTIETH.  75 

To  THE  Teacher. — The  reasoning  processes  of  the  following  exam- 
ples are  very  important,  and  should  be  thoroughly  understood  by  the 
scholars.  Not  till,  by  much  drill  and  many  repetitions,  they  have 
become  perfectly  familiar,  can  they  safely  be  omitted  or  neglected. 
Indeed,  if  the  pupil  must,  in  his  first  exercises,  omit  either,  it  is  far 
better  to  give  the  reasoning  process,  and  omit  the  answer,  than  to 
omit  the  process,  and  give  the  answer  only. 

1.  2  pks.  =  -jf  qts  ? 

Solution.  —  Since  1  peck  =  8  quarts,  2  pecks  must  equal  2  times  8 
quarts,  or  16  quarts.    Therefore,  2  pks.  =  16  qts. 

2.  6  qts.  =  ^  pts.  ?  5.     2  yds.  =  -x-  qrs. 

3.  2  wks.=  ^  da.  ?  G.     2  ,^.     ==  ^-  9  ? 

4.  2  dimes  =  -x-  cents?  7.     2  sq.  yds.  =  ^  sq.  ft.? 

8.  How  much  will  6  apples  cost  at  2  cents  a-piece  ? 

Solution.  —  Since  1  apple  costs  2  cents,  6  apples  will  cost  6  times  2 
cents,  or  12  cents.  Therefore,  6  apples  at  2  cents  each  will  cost  12 
cents. 

9.  How  much  will  8  books  cost  at  2  dollars  a-piece  ? 

10.  How  much  will  2  hats  cost  at  5  dollars  a-piece  ? 

11.  How  far  will  a  man  walk  in  2  hours,  if  he  walks  at  the  rate 
of  4  miles  per  hour  ? 

12.  How  many  bushels  will  8  boxes  hold,  if  each  box  holds  2 
bushels  ? 

13.  How  many  quarts  of  berries  will  George  pick  in  9  days,  if 
he  picks  2  quarts  per  day  ? 

14.  How  much  will  10  pairs  of  shoes  cost  at  2  dollars  per  pair? 

F.  1.     16  .^  =  *  §  ? 

Solution. — Since  8  drams  Apothecaries'  equal  one  ounce,  16  drams 
must  be  equal  to  as  many  ounces  as  there  are  times  8  in  16,  which  are 
2  times.     Therefore,  16  3  =  2  3. 


76  colburn's  first  part. 


2.  6  ft.    =  *  yds.  ?  5.     16  qts.  =  *  pts.  ? 

3.  20  pts.  ==  *  qts.  ?  6.       8  qrs.  =  -5^  yds.  ? 

4.  14  da.  =  *  wks.  ?  7.     16  pts.  =  *  qts.  ? 

8.  How  many  apples  at  2  cents  a-piece  can  be  bouglit  for  12 
cents  ? 

Solution.  —  If  one  apple  can  be  bought  for  2  cents,  as  many  apples 
can  be  bought  for  12  cents  as  there  are  times  2  in  12,  which  are  6 
times.  Therefore,  6  apples  at  2  cents  a-piece  can  be  bought  for  12 
cents. 

9.  How  many  oranges  at  3  cents  a-piece  can  be  bonght  for  6 
cents  ? 

10.  How  many  shawls,  at  $7  each,  can  be  bought  for  $14? 

11.  How  many  boxes,  holding  2  bushels  each,  will  be  required 
to  hold  20  bushels  of  apples  ? 

Solution.  —  If  1  box  is  required  to  hold  2  bushels,  as  many  boxes 
will  be  required  to  hold  20  bushels  as  there  are  times  2  in  20.  There- 
fore, 10  boxes,  each  holding  2  bushels,  will  be  required  to  contain  20 
bushels  of  apples. 

12.  How  many  hours  will  it  take  a  man  to  walk  14  miles,  if  he 
walk  at  the  rate  of  2  miles  per  hour  ? 

13.  How  many  days  would  it  take  a  man  to  earn  10  dollars,  if 
he  earned  2  dollars  per  day  ? 

14.  How  many  pieces  8  feet  in  length  can  be  cut  from  a  piece 
of  string  16  feet  in  length  ? 


LESSON    THIRTY-FIRST. 

77 

LESSON  XXXI. 

A. 

Add  10  threes  together. 

Note.—  The  pupil  should  supply  the  missing  part  of  this  and  the 
subsequent  tables,  by  his  knowledge  of  the  preceding  ones. 

TABLE. 

3  times  3  =  9. 

3  times  7,  or  7  times  3  = 

21. 

3  times  4,  or  4  times  3  =  12. 

3  times  8,  or  8  times  3  = 

24. 

3  times  5,  or  6  times  3  =  15. 

3  times  9,  or  9  times  3  = 

27. 

3  times  6,  or  6  times  3  =  18. 

3  times  10  or  10  times  3  = 

30. 

B.  1. 

*  times  3  =  15  ? 

5.       9  =  ^^  times  3  ? 

2. 

*  times  3  =    6  ? 

6.     12  =  4ftimes4? 

3. 

*  times  7  =  21  ? 

7.     U  =  ^  times  7  ? 

4. 

*  times  3  =  30  ? 

8.     24  =  *  times  8  ? 

C.  1. 

6  times  *  =  18? 

5.     12  =  3  times  *  ? 

2. 

3  times  *  =  21  ? 

6      24  =  3  times  *  ? 

3. 

3  times*  =  27? 

7.     15  =  5  times  ^  ? 

4. 

8  times  *  =  16  ? 

8.     30  ==  3  times  -^  ? 

D.  1. 

5  times  3,  plus  5  =  * 

times  2  ? 

2. 

3  times  8,  plus  6  =  * 

times  3  ? 

3. 

9  times  3,  minus  9  = 

*  times  6  ? 

4. 

4  times  3,  plus  4  =  * 

times  8  ? 

; 

5. 

2  times  9,  plus  6  =  * 

times  3  ? 

7* 

78  colburn's  first  part. 


E.  1.     3  wks.  =  *  da.  ?  4.     3  yds   3  qrs.  ==  *  qr.  ? 

2.  2  bu.  5  pks.  =  *  pks.  ?         6.     3  pks.  7  qts.  =  ^  qts.  ? 

3.  9  yds.  2  ft.  =  *  ft.  ?  6.     7  qts.  1  pt.  =  *  pts. 

7.  Francis  says  that  he  has  money  enough  to  buy  3  cocoa-nuts 
at  9  cents  a-piece,  and  still  have  6  cents  left.  How  much  money 
has  he  ? 

8.  William  has  9  three-cent  pieces,  and  8  cents  besides.  How 
many  cents  has  he  in  all  ? 

9.  Arthur  has  3  half-dimes,  and  2  three-cent  pieces.  How 
much  money  has  he  ? 

10.  How  many  pen-holders,  at  3  cents  a-piece,*can  be  bought 
for  15  cents  ? 

11  Willie  had  27  cents,  which  he  exchanged  for  their  value  in 
three-cent  pieces.     How  many  three-cent  pieces  did  he  get  ? 

12.  Amelia  had  20  very  nice  apples.  She  ate  2,  and  divided 
the  rest  among  her  playmates,  giving  3  to  each.  Among  how 
many  did  she  divide  them  ? 

13.  If  Augustus  has  37  apples,  how  many  will  he  have  left  after 
giving  3  of  his  companions  7  apples  a-piece  ? 

14.  Sarah  bought  9  spools  of  thread  at  3  cents  a-piece,  and 
then  had  money  enough  left  to  buy  2  skeins  of  silk  at  3  cents  per 
skein.     How  much  money  had  she  at  first  ? 

15.  Simon  had  42  cents.  He  gave  10  cents  for  a  writing  book, 
and  5  for  an  inkstand,  and  then  exchanged  the  rest  of  his  money 
for  three-cent  pieces.     How  many  three-cent  pieces  did  he  get  ? 


LESSON    THIRTY-SECOND.  79 


LESSON  XXXII. 

TABLE. 

A.     Add  10  fours  together. 

4  times  4  =  16.  4  times  8,  or  8  times  4  =  32. 

4  times  5,  or  6  times  4  =  20.  4  times  9,  or  9  times  4  =  36. 

4  times  6,  or  6  times  4  =  24.  4  times  10,  or  10  times  4=  40. 
4  times  7,  or  7  times  4  =  28. 

B.    Explanations  and  Definitions. — To  multiply  a  number  by  4, 
is  the  same  as  to  find  4  times  that  number;  to  multiply  a  number  by 
7  is  the  same  as  to  find  7  times  that  number,  <fcc.,  Ac.     Thus,  to  mul- 
tiply 6  by  4  is  the  same  as  to  find  4  times  6,  which  is  24. 
6  multiplied  by  3  =  3  times  5  =  15. 
8  multiplied  by  4  =  4  times  8  =  32. 
Multiplication,  theriy  is  the  process  of  finding  any  number  of  times 
a  given  number. 

The  number  to  be  taken  some  number  of  times  is  called  the  multi- 
plicand ;  the  number  showing  how  many  times  it  is  to  be  taken  is 
called  the  multiplier  ,•  the  answer  is  called  the  product. 

Thus,  in  "  8  times  3  =  24,"  8  is  the  multiplier,  3  the  multiplicand, 
and  24  is  the  product, 

Name  the  multiplier,  multiplicand,  and  product  in  each  of  the 
following  examples : — 

1.  9  times  4  =  36.  4.     4  times  6  =  24. 

2.  3  times  8  =  24,  6.     3  times  3  =  9. 

3.  7  times  4  s=r  28.  6.     4  times  4—16. 

The  multiplier  and  multiplicand  are  called  factors  of  the 
product. 

Thus,  in  9  times  4  =  36,  9  and  4  are  factors  of  36. 
Name  the  factors  in  the  above  examples. 


80 

colburn's  first  part. 

Two  oblique  lines  crossing  thus,  X  >  form  the  sign  of  multipli- 

cation. 

It  may  be  read  either  as  *'  times"  or  as  "  multiplied  by." 

Thus, 

"6X3  =  18"  may  be  read  either  as  "  6  times  3  =  18,"  or 

"  6  multiplied  by  3  =  18." 

To 

the  Teacher.  —  It  will  probably  be  well  to  have  the  pupils  at  1 1 

first 

reac 

the  sign  of  multiplication  as  though  written  "times;"  but 

they 

should  learn  to  read  and  use  it  in  either  way,  as  occasion  may  1 1 

require. 

c. 

1. 

*X4  =  16?                           9.     40  =  *x4? 

2. 

*  X  3  =  18  ?                         10.     28  =  *  X  4  ? 

3. 

*  X  4  =  36?                         11.     24  =  *  X  4? 

4. 

*X5  =  20?                         12.     32  =  *  X  8? 

5. 

4  X  *  =  40  ?                         13.     32  =  4  X  *  ? 

6. 

6x*  =  20?                         14.     36  =  4  X*? 

7. 

4x*  =  28?                         15.     24  =  6  X*? 

8. 

2x*  =  12?                         16.     16  =  4  X*? 

D. 

1. 

7  times  4,  plus  8  =  *  times  9  ? 

2. 

9  times  8,  plus  5  =  *  times  4  ? 

3. 

5  times  3,  plus  3  times  7  =  *  times  4  ? 

4. 

2  plus  7,  plus  6  times  3  =  -x-  times  8  ? 

E. 

1. 

3  pk.  6  qt.  =  *  times  5  qt.  ? 

Solution. — 3  pk.  6  qt.  =  30  qt. ;  and  30  qt.  contains  5  qt.  as  many 

times  as  30  contains  5,  which  are  6  times.   Hence  3  pk.  6  qt.  =  6  times 

5qts 

Abbreviated  Solution. — 3  pk.  6  qt.  =  30  qt. ;  and  30  qt.  =  6  times  1 1 

5  qt. 

Hence  3  pk.  6  qt.  =  6  times  5  qt. 

2. 

3  wk.  3  da.  =  *  times  4  da.  ? 

3. 

7  gal.  2  qt.  =  •}«•  times  3  qt.  ? 

4. 

2  wk.  4  da.  ==  *  times  6  da.  ? 

5. 

6^23=  times  2  9  ? 

6. 

8  yd.  =  ^t  times  1  yd.  1  ft.  ? 

7. 

4  gal.  2  qt.  =  ^  times  1  gal.  2  qt. 

LESSON    THIRTY-SECOND.  81 


8.  Edward  can  -walk  4  miles  per  hour,  and  Herma-n  can  walk 
3.     How  far  can  Edward  walk  in  6  hours  ?     Can  Herman  ? 

9.  Richard  bought  8  newspapers  at  2  cents  a-piece,  and  scld 
them  for  4  cents  a-piece.     How  much  did  he  gain  on  them  ? 

10.  If  Daniel  has  50  chestnuts,  how  many  wiU  he  have  left 
after  giving  4  of  his  companions  9  chestnuts  a-piece  ? 

11.  I  bought  9  yards  of  cloth  at  $4  per  yard,  but  it  being 
damaged,  I  was  obliged  to  sell  it  for  $12  less  than  it  cost  me. 
For  how  much  did  I  sell  it  ? 

12.  1  bushel  =  *  quarts? 

13.  1  yard     r=  -x-  nails  ? 

14.  4  pt.         =  ^  gills  ? 

15.  Arthur  had  a  basket  which  held  just  4  qt.,  and  he  picked 
nuts  enough  to  fill  it  6  times.  How  many  quarts  did  he  pick  ? 
How  many  pecks  ? 

16.  How  many  oranges  at  4  cents  each  can  be  bought  for  6 
three-cent  pieces  and  2  cents  ? 

17.  How  many  apples  at  3  cents  a-piece  can  be  bought  for  6 
oranges  at  4  cents  each  ? 

18.  How  many  pairs  of  boots  at  $5  a  pair,  can  be  bought  for 
10  yards  of  cloth  at  $3  per  yard  ? 

19.  A  man  bought  10  quarts  of  berries,  which  he  put  into  boxes 
each  holding  5  pints.  How  many  pints  of  berries  did  he  buy  ? 
How  many  boxes  did  he  fill  ? 

20.  Lucius  is  shelling  corn  into  a  three-peck  measure,  which  he 
empties  into  a  bin  large  enough  to  hold  6  bushels  3  pecks.     How 
many  pecks  must  he  shell  to  fill  the  bin  ?     How  many  measure-  '  I 
fuls? 

21.  A  newsboy  sold  9  papers  at  8  cents  a-piece,  and  after 
spending  9  cents,  gave  the  rest  of  his  money  for  papers  at  2  cents 
a-piece.     How  many  papers  did  he  get  ? 


82  COLBURN*S    FIRST     PART. 


22.  A  man  gave  8  hats  at  $4  a-piece,  and  $8  in  money  for  coats 
at  $10  a-piece.     How  many  coats  did  he  receive  ? 

23.  A  man  put  7  gallons  2  quarts  of  molasses  into  jugs  each 
holding  3  quarts.     How  many  jugs  did  he  fill  ? 

24.  Rufus  had  a  string  5  yards  1  foot  long,  which  he  cut  into 
pieces  just  2  feet  long.     How  many  pieces  did  it  make  ? 

25.  An  apothecary  put  6  .^  1  9  of  powders  into  papers,  each 
holding  2  9.     How  many  papers  did  he  fill  ? 


LESSON  XXXIII. 

A.  Add  10  fives  together. 

5  times  5  =  25. 

6  times  6,  or  6  times  5  =  30. 
6  times  7,  or  7  times  6  =  35. 
6  times  8,  or  8  times  6  =  40. 
6  times  9,  or  9  times  5  =  45. 

5  times  10,  or  10  times  5  =  50. 

B.  Add  10  sixes  together. 

6  times  6  =  36. 

6  times  7,  or  7  times  6  ==  42. 
6  times  8,  or  8  times  6  =  48. 
6  times  9,  or  9  times  6  =  54. 
6  times  10,  or  10  times  6  =  60. 

CI.  *X6==36?  6.  42=r*x7? 

2.  *X8  =  48?  6.  54  =  *x9? 

8.  *x5  =  45?  7.  54  =  *x6? 

4.  *X6  =  30?  8.  25=:*x5? 


LESSON    THIRTY-THIRD.  83 


D.  1.  9x*  =  54?  6.  60  =  6  X*? 

2.  7X*=35?  6.  40  =  8  X*? 

3.  6  X  ^  =  48?  7.  86  =  6  X  *? 

4.  8X*  =  40?  8.  42  =  7  X*? 

Explanation. — To  diride  a  nnmber  by  2  is  to  find  how  many  times 

2  equal  it. 

To  divide  a  number  by  6  is  to  find  how  many  times  6  equal  it 
Hence,  35  divided  by  5  =  7i  for  35  =  7  times  5;    48  divided  by 

8  =  6,  for  48  =  6  times  8. 

Division,  then,  is  the  process  of  finding  how  many  times  one  number 
must  be  taken  to  equal  another  number 

The  number  to  bo  divided  is  called  the  dividend.  The  number  br 
Trhich  we  divide  is  called  the  divisor,  and  the  answer  is  called  the 

QUOTIENT. 

Thus,  in  35  divided  by  7  ==  5,  35  is  the  dividend,  7  is  the  divisor, 
5  is  the  quotient. 

In  54  =»  *  times  6,  54  is  the  dividend,  &  is  the  divisor,  and  9,  the  an- 
swer, is  the  quotient. 

Name  the  divisor,  dividend,  and  quotient  of  the  following 
examples :  — 

28  divided  by  7  =  4.  86  =  *  x  6  ? 

48  divided  by  6  =  8.  20  =  *  x  5  ? 

85  divided  by  7  =  5.  28  =  *  X  7  ? 

By  these  illustrations  it  appears  that  division  is  just  the  reverse 
of  multiplication. 

A  horizontal  line  with  one  dot  above,  and  another  below  it, 
forms  the  sign  of  division,  thus :  —.    It  may  be  read  *'  divided  by." 
28  -r  7  =  4  may  be  read,  "28  divided  by  7  equal  4." 

F.  What  is  the  quotient  of  — 

1.  16  ~- 4?             4.     25  —  5?  7.  18-^3? 

2.  24—3?              6.     48-7-6?  8.  24—8? 

3.  32  -f-  8  ?              6.     42  -T-  7  ?  9.  64  -7-  9  ? 


84  colburn's   first   part. 


Ct.  1.     5  wk.  Id.  -f-  1  wk.  2  da.  ? 

Solution.  —  5  wk.  1  da.  —  36  daj'^s ;  1  wk.  2  da.  =  9  days  j  and  36 
da.  =  4  times  9  da.     Hence,  5  wk.  1  da.  -r  1  wk.  2  d.  =  4. 

2.  3  pk.  6  qt.  -r-  1  pk.  2  qt. 

3.  6  yd.  2  ft.    -^  1  yd.  2  ft. 

4.  9  gal.  -i-  2  gal.  1  qt. 

5.  6  wk.  3  da.  -r  1  wk.  2  da. 

6.  3g63-M§23. 

7.  3  sq.  yd.  6  sq.  ft.  ~  8  sq.  ft. 

8.  I  sold  7  quarts  of  chemes  at  6  cents  per  quart,  and  one 
quart  for  8  cents.  How  much  did  I  receive  ?  I  expended  the 
money  thas  received  for  rice  at  5  cents  per  pound.  How  many 
pounds  of  rice  did  I  buy  ? 

9.  If  George  walks  at  the  rate  of  15  rods  per  minute,  and  Wil- 
liam walks  at  the  rate  of  21  rods  per  minute,  how  many  more  rods 
per  minute  does  William  walk  than  George  ?  How  many  more 
rods  in  9  minutes. 

10.  If  Susan  gains  8  merit-marks  per  day,  and  loses  2  per  day, 
how  many  will  she  have  at  the  end  of  8  days  ? 

11.  A  man  sold  5  pecks  of  chestnuts  at  the  ratie  of  one  dime 
per  quart.     How  many  dimes  did  he  receive  ? 

12.  I  sold  6  quarts  of  blackberries  at  the  rate  of  10  cents  per 
quart,  and  received  in  payment  5  three-cent  pieces,  and  the  rest 
in  half-dimes.     How  many  half-dimes  did  I  receive  ? 

13.  Abner  and  Lemuel  were  in  a  store  together,  and  their  father 
tt)ld  them  that  they  might  each  have  8  oranges  worth  6  cents  a- 
piece,  or  9  worth  5  cents  a-piece.  Abner  chose  the  former,  and 
Lemuel  the  latter.  How  much  more  were  Abner's  oranges  worth 
than  Lemuel's  ? 

14.  How  many  bags,  each  containing  1  bu.  1  pk.,  can  be  filled 
from  8  bu.  3  pk.  of  meal  ? 


LESSON  THIRTY- FOURTH. 


15.  How  many  house-lots,  each  containing  1  A.  2  R.,  can  be 
made  from  a  piece  of  land  containing  10  A.  2  R.  ? 

16.  How  many  pictures  at  2d.  1  qr.  each  can  be  purchased  for 
6d.  3  qr.  ? 

17.  How  many  bushels  in  8  bags,  each  containing  3  pecks  ? 

18.  A  furniture-dealer  gave  6  bureaus,  worth  7  dollars  a-piece, 
and  3  dollars  in  money,  for  chairs  at  9  dollars  per  dozen.  How 
many  dozen  chairs  did  he  buy  ? 

19.  A  fur-dealer  gave  8  caps,  at  5  dollars  a-piece,  and  2  dollars 
in  money,  for  muffs  at  G  dollars  a  piece.  How  many  muffs  did  he 
receive  ? 

20.  A  boy  earned  12  cents  by  doing  some  errands,  and  invested 
the  money  in  papers  at  2  cents  a-piece.  He  sold  the  papers  at 
4  cents  each,  and  with  the  money  received  for  them  he  bought 
papers  at  3  cents  a-piece.  He  sold  C  of  the  papers  for  5  cents 
a-piece,  and  the  rest  for  2  cents  a-piecc.  He  then  spent  4  cents 
for  crackers,  and  gave  the  rest  of  his  money  for  some  very  nice 
Havana  oranges  at  6  cents  a-piece.  He  gave  1  of  the  oranges  to 
his  mother,  and  sold  the  rest  at  8  cents  a-piece.  How  much  did 
he  receive  for  them  ? 


LESSON  XXXIV. 

7  times  7  =  49. 

7  times  8,  or  8  times  7  =  56. 

7  times  9,  or  9  times  7  =  63. 

7  times  10,  or  10  times  7  =  70. 

8  times  8  =  64. 

8  times  9,  or  9  times  8  =  72. 
8  times  10,  or  10  times  8=  80. 


86  COLBURN*S    FIRST    PART. 

9  times  9  =  81. 

9  times  10,  or  10  times  9  =  90. 

10  times  10  =  100. 

B.  66==*x8?  90  =  *xl0? 
81  =  *  X  9?                                    49_  ^e  ><  7^ 

63  =  ^X7?  100=*xlO? 

64  =  ^X8?  72  =  *x8? 

C.  81-7-9?  56-^8?  100—10? 
36-7-4?  21 -j- 7?                      24 -r  8? 
48 -i- 8?  45 -f- 6?                      72  H- 9? 
72 -f- 9?  81-7-9?                      63-7-7? 

D.  1.  7  times  8,  plus  7,  divided  by  7,  multiplied  by  4  =  -k- 
times  6  ? 

Solution. — 7  times  8  =  56,  plus  7  =«  63,  divided  by  7  =*  9,  multi- 
plied by  4  ==  36,  which  equals  6  times  6. 

Note. — Let  the  pupil  learn  to  call  only  the  results,  thus  :  56,  63, 
9,  36,  6.  Let  them  also  perform  the  work  as  the  Teacher  reads  the 
example. 

2.  8  times  10,  minus  8,  divided  by  9,  plus  1,  multiplied  by  6, 
minus  5  =  *  times  7  ? 

3.  6  times  8,  minus  12,  divided  by  4,  multiplied  by  3,  plus  10 
times  4,  minus  3  =  *  times  8  ? 

4.  7  times  6,  plus  3,  divided  by  9,  multiplied  by  5,  plus  3, 
divided  by  7,  multiplied  by  6,  minus  2  =  *  times  2  ? 

5.  3  times  8,  plus  4  times  9,  plus  21,  divided  by  9,  multiplied 
by  2,  plus  7,  divided  by  5  ? 

E.  1.     How  many  are  9  times  2  yds.  2  ft.  ? 

Solution. —  9  times  2  yd.  =  18  yd.;  9  times  2  ft.  ==  18  ft.  =-  6 
yds.,  which,  added  to  18  yd.  =  24  yd.     Hence,  9  times  2  yd.  2  ft. 
24  yd. 


LESSON    THIRTY-FOURTH.  87 

2.  4  times  7  gal.  3  qt.  ?  6.     9  times  3  sq.  yd.  4  sq.  ft.  ? 

3.  7  times  8  wk.  4  da.  ?  6.     6  times  7  bu.  6  pk.  ? 

4.  6  times  9  yd.  2  ft.  ?  7.     6  times  8  gal.  2  qt.  ? 

8.  Bought  6  bags,  each  containing  8  bu.  2  pk.  of  peanuts,  and 
put  them  into  casks  each  holding  3  bushels.  How  many  casks 
did  they  fill? 

9.  Austin  had  a  basket  which  held  just  3  pk.  4  qt.,  and  an- 
other which  held  just  6  pecks.  He  gathered  nuts  enough  to  fill 
the  smallest  basket  10  times.  How  many  times  would  they  fill 
the  large  basket? 

10.  A  lady  bought  cotton  sheeting  enough  to  make  8  sheets, 
each  containing  6  yd.  1  qr.,  but  afterwards  concluded  to  put  6 
yards  in  a  sheet.     How  many  sheets  could  she  make  ? 

11.  I  bought  a  vessel  of  oil  containing  32  quarts,  and  after 
using  1  gal.  3  qts.  of  it,  I  sold  the  rest  at  the  rate  of  1  dollar  for 
1  gal.  1  qt.     How  much  did  I  get  for  it  ? 

12.  Sarah's  mother  oflfered  to  give  her  8  large  oranges  worth  5 
cents  a-piece  if  she  would  tell  her  how  many  seven-quart  baskets 
could  be  filled  from  5  pk.  2  qt.  of  berries.  Sarah  answered  cor- 
rectly. What  was  her  answer  ?  She  exchanged  the  oranges  for 
their  value  in  drawing-pencils  worth  10  cents  a-piece.  How  many 
pencils  did  she  get  ? 

13.  A  fruit-peddler  paid  48  cents  for  cherries  at  6  cents  a  quart, 
which  he  put  into  papers,  each  containing  1  gill.  How  many 
papers  did  it  take  ? 

14.  By  bujdng  a  lot  of  wood  at  $4  per  cord,  and  selling  it  at  $6 
per  cord,  I  gained  $18.  How  many  dollars  did  I  gain  on  each 
cord  ?  How  many  cords  did  I  buy  ?  How  many  dollars  did  I 
pay  for  the  entire  lot  ? 

15.  By  buying  flour  at  $5  per  barrel,  and  selling  it  for  $8  per 


88  colburn's  first  part. 


barrel,  I  gained  $24.    How  many  barrels  did  I  buy,  and  what  did 

1  pay  for  the  lot  ? 

16.  By  buying  a  lot  of  cloth  for  $5  per  yard,  and  selling  it  at 
$3  per  yard,  I  lost  $20.  IIow  many  dollars  did  I  pay  for  the 
lot? 

17.  A  man  bought  a  lot  of  coal  at  $4  per  ton,  and  sold  it  for 
$8  per  ton,  by  which  he  gained  $36.  How  many  dollars  did  he 
pay  for  it  ? 

18.  A  laborer  worked  6  weeks  for  9  dollars  per  week,  and  put- 
ting 6  dollars  with  the  money  thus  earned,  he  bought  coal  at  6 
dollars  per  ton.     How  many  tons  did  he  buy  ?   After  laying  aside 

2  tons  for  his  own  use,  he  sold  the  remainder  for  7  dollars  per 
ton,  receiving  in  payment  2  dollars  in  money,  and  the  rest  in 
flour  at  9  dollars  per  barrel.  How  many  ban*els  of  flour  did  he 
receive  ? 

19.  Robert  had  a  basket  holding  2  quarts  1  pint,  and  he 
gathered  chestnuts  enough  to  fill  it  8  times.  How  many  four- 
quart  baskets  could  he  fill  with  what  he  gathered  ? 

20.  Augustus  had  8  quarts  of  blackberries.  He  sold  1  quart 
for  11  cents,  and  the  rest  for  10  cents  per  quart.  With  the  money 
received  for  them  he  bought  cocoa-nuts  at  9  cents  a-piece.  How 
many  cocoa-nuts  did  he  buy  ?  He  divided  2  of  them  among  his 
companions,  and  sold  the  rest  for  10  cents  a-piece.  He  gave  7 
cents  to  a  poor  woman,  and  spent  the  rest  of  his  money  for  fire- 
crackers, at  7  cents  a  bunch.     How  many  bunches  did  he  buy  ? 


LESSON    THIRTY-FIFTH.  89 


LESSON  XXXV. 

BiUs. 

A.  1.  Mr.  Edward  Crane  keeps  a  store  in  Boston.  On  the 
21st  of  February,  1856,  lie  sold  to  Mr.  Alfred  Hall  7  yards  of 
broadcloth  at  $5  per  yard,  9  yards  of  cassimere  at  $2  per  yard, 
and  8  yards  of  doeskin  at  $3  per  yard.  Mr.  Hall  paid  for  them, 
and  asked  Mr.  Crane  to  make  out  a  hillf  i.  e.j  a  written  statement 
of  the  transaction.     He  did  it  as  follows  : — 

y  yc/^j.,  SSzoac/ciomj  a  ^5       .       .       .     ^35 
P  "^ad,   %addime^ej    a  j&S  .       .  ^o 


J^77 


If  the  goods  had  not  been  paid  for,  Mr.  Crane  would  not  have 
receipted  the  bill,  i.  «.,  he  would  not  have  put  his  name  after  the 
words  "received  payment." 

Note  to  the  Teacher.  —  A  more  full  explanation  may  be  found 
in  "Arithmetic  and  its  Applications." 


90  colburn's  first  part. 


B.  Make  out  proper  bills  for  each  of  the  following  transactions, 
observing  to  give :  — 

1st.  The  date,  t.  «.,  the  place  and  time. 
2d.  The  names  of  the  parties. 

3d.  The  articles  bought,  with  their  prices  and  amount. 
4th.  The  words  **  received  payment,** 

6th.  1/  the  good*  art  paid  for ,  or  bought  for  cash,  the  name  of  the 
seller. 

Note. — It  may  add  to  the  interest  and  value  of  the  following  pro- 
blems, to  have  each  pupil  write  his  own  name,  and  that  of  some  com- 
panion, in  the  place  of  those  here  written. 

1.  John  Brown,  of  New  York,  sold  to  Martin  Draper,  for  cash, 
July  16th,  1856,  9  bbls.  of  flour  at  $10  per  barrel,  and  8  bushels 
of  wheat  at  $2  per  bushel. 

2.  William  Fuller,  of  Chicago,  sold  to  Ai-thur  Simmons  for  cash, 
Jan.  8th,  1856,  9  silk  hats  at  $4  a-piece,  8  cloth  caps  at  $2 
a-piece,  6  beaver  hats  at  $5  a-piece,  and  11  di-ab  hats  at  $1 
a-piece. 

3.  Henry  Mitch  el  &  Co.  sold  to  Francis  Baker,  on  credit  (t.  «., 
Mr.  Baker  at  the  time  did  not  pay  for  them),  May  1st,  1855,  6 
bbls.  of  apples  at  $3  per  barrel,  9  bbls.  of  potatoes  at  $2  per 
barrel,  7  boxes  of  raisins  at  $3  per  box,  and  5  drums  of  figs  at 
$2  per  drum. 

Note.  —  Let  the  students  now  make  out  bills  for  several  imaginary 
transactions. 

C.  If  in  the  transaction  described  under  letter  A.,  Mr.  Hall  had 
paid  $2  in  money  and  delivered  to  Mr.  Crane  3  coi*ds  of  wood  at 
$9  per  cord,  the  bill  would  have  been  made  out  as  follows : 


LESSON    THIRTY-FIFTH, 

91 

^o.^lon,  c%/  i^/. 

*^B56. 

.^35 

0  yc/d.   S)o6d4cro^    a  p3 

.     SA 

^77 

i'                       9;. 

i     ^   "^a^A 

ps 

.       2A 

^36 

/4/ 

Make  out  a  bill  for  the  following  transaction : — 

1.  Moses  White,  a  trader  of  Philadelphia,  sold  to  Joseph  Aus- 
tin, May  17th,  1856,  7  ploughs  at  $9  each,  5  hay-cutters  at  $8 
a-piece,  4  doz.  scythes  at  $9  per  dozen,  and  8  doz.  rakes  at  $3 
per  dozen ;  and  in  part  payment,  Mr.  Austin  gave  him  8  cords  of 
wood  at  $7  per  cord,  4  cords  of  wood  at  $6  per  cord,  and  $13  in 

money. 

Note. — Ld-t  thf  scholars  make  out  bills  for  several  imaginary  trans- 
*'^tions. 

.'    .: • . 

92 


COLBURN    S     FIRST     PART. 


LESSON  XXXVI. 

A.  1.     29  =  *  times  6  ? 

First  Solution.— 29  =  24  +  5^  and  24  =  4  times  6.  Hence  29 
=  4  times  6  with  5  remainder. 

Second  Solution. — 29  =  4  times  6  with  5  remaining,  for  4  times 
6  =  24,  and  5  =  29. 

Note. — The  remainder  is  really  an  undivided  part  of  the  dividend, 
and  might  be  subtracted  from  it  without  affecting  the  quotient.  In 
reality,  but  a  part  of  the  dividend  is  divided. 


2.  46  =  *  times  5  ? 

3.  31  =  ^  times  9  ? 

4.  67  =  *  times  7  ? 

5.  88  =  -Jf  times  9  ? 

6.  61  =  *  times  8  ? 

7.  37  =  *  times  7  ? 

B.  1.  42  —  4? 

2.  83-^9? 

3.  75  -^  8  ? 

4.  24 -f- 7? 


13. 

6.  19  ~  8  ? 

6.  47—7? 

7.  39  ~-  6  ? 

8.  51-^8? 


8.  63  =  *  times  6  ? 

9.  27  =  *  times  8? 

10.  62  =  #  times  9  ? 

11.  48  =  -5^  times  7  ? 

12.  69  =  *  times  9  ? 
23  =  *  times  11  ? 


9.  43 -f- 9? 

10.  43  -r-  4  ? 

11.  47 -r  8? 

12.  47  -T-  6  ? 


C.  The  comma,  when  used  in  connexion  with  Arithmetical  signs, 
shows  that  the  result  of  all  the  preceding  operations  is  to  be  con- 
sidered in  connexion  with  the  sign  following  it. 

Thus :  "4  X  9,  -r-  7,"  means  that  4  times  9,  or  36,  is  to  be  divided 
by  7.  I 

**6  X  7  4-  18,  -7-  9,"  means  that  the  sum  of  6  times  7  plus  18,  »=  1 
42  -|-  18  =  60,  is  to  be  divided  by  9. 


LESSON    THIRTY-SIXTH.  93 


1.  6x9,-8?  5.  9x6+10,  =  *X  8? 

2.  4x6, -T-T?  6.  4x9+17,  =  *X  6? 

3.  5x9  +  8,-7-6?  7.  7  X  7  + 13,  =  *  X  10? 

4.  8  X  8  —  13,  -r  9  ?  8.  9  X  9  --  43,  =  *  X  4  ? 

D.  1.     41  da.  =  *  wk.? 

Solution. — Since  7  days  =  1  wk.,  41  da.  must  equal  as  many  wk. 
as  there  are  times  7  in  41,  which  are  5  times  with  6  remainder.  Hence 
41  da.  =  5  wk.  6  da. 


2. 

33  qr.  ==  *  yd.  ? 

7. 

46  fur.  =  *  m.  ? 

3. 

37,:^     =^§? 

8. 

69  da.    =  *  wk.  ? 

4. 

70  qt.  =  *  pk.  ? 

9. 

37  cd.  ft.  =  *  cd.  ? 

5. 

19  ft.    =  *  yd.  ? 

10. 

60  sq.  ft.  =  *  sq.  yd.  ? 

6. 

23  ft.    =  ^  yd.  ? 

11. 

20  m.  =  *  le.  ? 

12.  Moses  has  35  cents,  with  which  he  wishes  to  buy  oranges 
at  6  cents  a-piece.     How  many  oranges  can  he  buy  ? 

Solution. — If  he  can  buy  1  orange  for  6  cents,  he  can  buy  as  many 
oranges  for  35  cents  as  there  are  times  6  in  35,  which  are  5  times  and 
5  remainder.  Therefore,  he  can  buy  5  oranges,  and  have  5  cents 
remaining. 

13.  How  many  barrels  of  flour,  at  $6  per  baiTel,  can  be  bought 
for  $58  ? 

14.  How  many  shawls,  at  $7  a-piece,  can  be  bought  for  $39  ? 
16    How  many  books,  at  $3  a-piece,  can  be  bought  for  $29  ? 

16.  How  many  terms  tuition,  at  $7  per  term,  will  $27  pay  for? 

17.  How  many  hats,  at  $4  each,  can  be  bought  for  $39? 

18.  How  many  cans,  each  holding  1  gal.  1  qt.,  can  be  filled 
from  3  gal.  3  qt.  of  milk? 


94  colburn's  first  part. 


19.  A  person  who  owes  $39,  wishes  to  pay  as  much  as  possible 
m  five-dollar  bills,  and  the  rest  in  one-dollar  bills.  How  many 
bills  of  each  kind  must  he  pay  ? 

20.  William  did  8  errands,  for  each  of  which  he  received  3 
cents,  and  he  had  11  cents  before  he  did  the  errands.  He  bought 
as  many  writing-books  at  9  cents  a-piece  as  he  could  pay  for,  and 
spent  the  rest  of  his  money  for  pens  at  2  cents  a-piece.  How 
many  writing-books  did  he  buy  ?     How  many  pens? 

21.  A  tailor  paid  $35  for  silk  velvet  at  $5  per  yard.  He  made 
it  into  vests,  putting  3  quarters  into  each  vest.  How  many  vests 
did  he  make,  and  how  many  quarters  had  he  remaining  ? 

22.  One  "  Fourth  of  July"  Thomas  had  29  cents.  He  bought 
as  many  bunches 'of  crackers  at  10  cents  per  bunch,  as  he  could 
pay  for,  and  then  spent  the  rest  of  his  money  for  cherries,  at  the 
rate  of  7  for  a  cent  How  many  bunches  of  crackers  did  he  buy  ? 
How  many  cherries  ? 

23.  A  person  who  had  20  cents,  said  to  a  boy:  **If  you  will 
tell  me  how  many  loaves  of  bread,  at  6  cents  per  loaf,  I  can  buy 
with  my  money,  I  will  give  you  what  there  is  left  after  paying  for 
the  bread."  The  boy  answered  right.  What  was  his  answer? 
How  many  cents  ought  the  person  to  give  him  ? 

21.  If  it  requires  3  yards  of  broadcloth  to  make  a  coat,  how 
many  coats  can  be  made  from  a  piece  containing  29  yards  of 
broadcloth  ?  How  many  yards  will  be  left  after  making  the  coats  ? 
If  one  yard  of  cloth  will  make  3  vests,  how  many  vests  can  be 
made  from  what  remains,  after  making  the  coats  ? 

25.  Lyman  has  28  cents,  Horace  has  50.  Chester  has  63,  and 
Isaac  has  47.  Each  bought  as  many  pencils  at  8  cents  a-piece  as 
he  could  pay  for,  and  gave  the  rest  of  his  money  to  a  poor  wo- 
man. How  many  pencils  did  each  buy,  and  how  many  cents  had 
each  to  give  the  poor  woman  ?  How  many  pencils  were  bought 
in  all  ?     How  many  cents  did  the  woman  receive  ? 


I  -  I 


LESSON 

THIRTY-SEVENTH 

95 

LESSON  XXXVII 

A.  1.     2  times  3  tens 

?     Then  8  times  30  ? 

2.     6  times  7  tens 

?     Then  5  times  70  ? 

3.     8  times  4  tens  ?     Then  8  times  40  ? 

4.     7  times  3  tens  ?     Then  7  times  30  ? 

B.  1, 

4X8? 

4. 

7x9? 

7. 

8X7? 

2 

4x  30? 

6. 

7X90? 

8. 

8x70? 

3. 

40x3? 

6. 

70  X  9? 

9. 

80  X  7? 

C    1. 

6  X  40? 

5. 

6x  30? 

9. 

9x  60? 

o 

9  X  30? 

6. 

60  X  3? 

10. 

8x  90? 

3. 

6x  40? 

7. 

9  X  70? 

11. 

7  X  70? 

4. 

40  X  4? 

8. 

90  X  7? 

12. 

8  X  80? 

D.  1. 

12- 

-4? 

4. 

64  ~  8? 

7. 

54 -r-  9? 

2. 

120  H 

-4? 

5. 

640 -f-  8? 

8. 

640  -T-  90  ? 

3. 

120- 

-  40? 

6. 

640 -r  80? 

9. 

540 -r  9? 

E.   1. 

630- 

r  9? 

8.     420- 

^60? 

2. 

720- 

-8? 

9.       90- 

-  30? 

3. 

560- 

-7? 

10.     270- 

-  30? 

4. 

250  H 

-6? 

11.    420- 

-  6? 

6. 

240- 

-  3? 

12.     810- 

-  9? 

6. 

720- 

-  80? 

13.     810  - 

-  90? 

7. 

180- 

r  3? 

14.     360  - 

-40? 

F.  1.     6  X  47? 

S0LUTION.—6  times  40  =  240 ; 
=  282.     Therefore,  6  times  47  = 

6  times  7  =  42 

=  282, 

,  which, 

added  to  240 

r^^ ^ 

,96  colburn's  iirst   part. 


Note.  —  As  soon  as  the  principle  is  understood,  the  pupil  should 
solve  such  problems  by  naming  only  the  results.  Thus:  6  times 
47  =  240  4-  42  =  282. 

2.  7  X  96  ?  7.  4  X  27  ?  12.  3  x  37  ? 

3.  8  X  34?  8.  9  X  82?  13.  6  x  28  ? 

4.  9  X  37  ?  9.  6  X  97  ?  14.  6  X  43  ? 

5.  6x94?  10.  4x23?  15.  9x81? 
G.  8x23?  11.  7x94?  16.  7x63? 

G.  "When  we  wish  to  write  the  work,  we  may,  if  we  choose, 
solve  exaipples  in  multiplication,  as  explained  in  the  following 
solution  of  the  first  question  under  F. 


A7 


Explanation. — 6  times  7  units  =-=  42  units  =  4  tens, 
and  2  units.  "Writing  the  2  units,  and  reserving  the  4  tens 
A  to  add  to  the  product  of  the  tens,  we  have  6  times  4  tens= 

24  tens,  to  which,  adding  the  4  tens  from  the  former  pro- 
duet,  gives  28  tens,  which  wo  write.  The  answer,  then, 
is  282. 


Note. —  It  will  be  seen  that  when  we  do  not  write  the  work,  we 
begin  at  the  left  hand  to  multiply;  and  when  we  do  write  it,  we  begin 
at  the  right  hand. 

Perform  in  this  way  the  examples  under  letter  F. 

H.  1.     6  times  498  ? 

Solution.  —  6  times  8  units  =  48  units  ==  4  tens  and  8  units. 
Writing  the  8  units,  and  reserving  the  4  tens  to  add  to  the  product  of 
the  tens,  we  have  6  times  9  tens  =  54  tens,  and  4  tens  added,  equal 
58  tens  =  5  hundreds  and  8  tens.  "Writing  8 
tens,  and  reserving  the  5  hundreds  to  add  to 
the  product  of  the  hundreds*  column,  we  have 
6  times  4  hundreds  =  24  hundreds,  and  5  hun- 
dreds added  =  29  hundreds  ==  2  thousands 
^^  and  9  hundreds,  which  being  the  last  product 

we  write,  the  answer  then  is  2988. 


^j?^ 


^ 


^0 


LESSON    THIRTY-EIGHTH.                  97 

Note. — When  the  reductions  are  fully  mastered,  abbreviated  forms 

like  the  following  may  be  introduced  with  advantage. 

6  times  8  =  48.     Write  8,  and  add  4  to  the  next  product.     6  times 

9  are  54  and  4  are  58. 
4  are  24  and  5  are  29, 

W^rite  8  and  add  5  to  the  next  product.    6  times 
which  we  write.     Hence,  6  times  498  »=-  2988. 

The  following  form  of  naming  only  results   should  finally  be 

adopted.     48  units ;  64,  58  tens ;  24,  29  hundreds,    ^n*.-— 2988. 

2.     9  X  847  ? 

8. 

6  times  $2.75? 

8.     8  X  298  ? 

9. 

4  times  $8.76? 

4.     4x746? 

10. 

9  times  $32.75? 

6.     8  X  327  ? 

11. 

8  times  $27.84  ? 

6.     4  X  238  ? 

12. 

5  times  $97.83  ? 

7.     6  X  379  ? 

13. 

4  times  $28.59  ? 

LESSON  XXXVIII. 

A.  1.     476  ^ 

7? 

Solution. — 7  is  contained  in  47  tens,  6  tens* 

times,  with  5  teni  re- 

maining.     5  tens  «=  50 

units,  and  6  units  added,  are  56  units,  7  is 

contained  in  56  units  8 

units'  times.    Hence,  476  -r  7  ««  60  -f-  8  — «  68. 

Note. — Not  till  the  reductions  are  fully  understood,  should  the 
pupil  be  allowed  to  abbreviate  this  explanation  to  the  common  one :  *'  7 
is  contained  in  47,  6  times,  with  5  remainder.    7  is  contained  in  56,  8 
times.    Hence  the  quotient  is  68." 

2.     315  —  8? 

6.     672 -^  8? 

10.     429  -7-  8  ? 

8.     216  —  4? 

7.     144 -H  6? 

11.     413  —  5? 

4.     392  -.  8  ? 

8.     279  -~  9  ? 

12.     673—7? 

5.     217—7? 

9.     137-^2? 

13.     528  ~-  9  ? 

98 


COLBtlRN    S    FIRST    PART. 


Examples  in  division  are  performed  and  explained  in  the  same 
manner  when  we  write  the  work  as  when  we  do  not.  The  work 
of  the  first  example,  letter  A.,  would  usually  be  written  as  in  the 
annexed  model : 


7y76 


6B 


B.  1.    2738  -4-  8  ? 


Solution. — 8  is  contained  in  27  hundreds,  3  hundred  times  with  3 
hundreds  remaining.     "VVe  therefore  write  3  as  the  hundreds*  figure 
'  Q  Q  ^  of  the  quotient.     The  3  hundreds  re 


})27t 


-2 


maining  =  30  tens,  and  3  tens  added 

r=  33  tens.    8  is  contained  in  33  tens, 

*^  //Q)  4r  tens  times,  with  1  ten   remaining. 

We  therefore  write  4  as  the  tens  figure 
of  the  quotient.  The  1  ten  remaining  =  10  units,  and  8  units  added  = 
IS  units.  8  is  contained  in  18  units  2  units'  times  and  2  units  re- 
maining. Therefore,  2738  -*-  8  ==  3  hundreds,  4  tens,  and  2  units,  or 
342  with  a  remainder  of  2. 

Note. — The  remainder  is  written  after  the  quotient  with  the  sign 
of  subtraction,  to  show  that  it  is  an  undivided  part  of  the  dividend. 


2.  4756  -T-  4  ? 

8.  3297  -7-  6  ? 
4.  4347-7-9? 
6.  2981  ^  11  ? 

6.  3297  -r  6  ? 

7.  '43G1  -^  5  ? 
a  2459  -f  8  ? 

9.  4272  ~  12  ? 
10.  8943  ^  9  ? 


11. 

2137- 

-5? 

12. 

4264  H 

-  13? 

13. 

8375- 

-  12? 

14. 

2986- 

r  4? 

15. 

3176  H 

-  8? 

16. 

4327  H 

-  9? 

17. 

2052- 

-  3? 

18. 

1379- 

-2? 

19. 

7436- 

-  8? 

LESSON    THIRTY-NINTH.                   99 

LESSON 

XXXIX. 

The  following  tables,  if  thoroughly  learned,  will  save  a  yast 

deal  of  labor  in  the  Arithmetical  operations  of  life.     A  distin- 

guished educator, 

now  Superintendent  of  Schools  in  one  of  the 

principal  cities  of  the  Union,  says 

that  in  his  opinion,  a  knowledge 

of  these  tables  would  save  hours  of  valuable  time,  not  only  to  the  1 1 

student. 

but  to  the  business  man. 

With  most  classes,  the  Teacher 

will  find  it  desirable  to  give  additional  exercises  similar  to  those  1 1 

of  the  preceding  Lessons. 

11  X 

2,  or  2  X 

11  =  22. 

16  X  2,  or  2  X  16  =  32. 

12  X 

2,  or  2  X 

12  ==  24. 

17  X  2,  or  2  X  17  =  34. 

13  X 

2,  or  2  X 

13  =  26. 

18  X  2,  or  2  X  18  =  36. 

14  X 

2,  or  2  X 

14  =  28. 

19  X  2,  or  2  X  19  =  38. 

15  X 

2,  or  2  X 

15  ==  30. 

20  X  2,  or  2  X  20  =  40. 

11  X 

3,  or  3  X 

11  =  33. 

16  X  3,  or  3  X  16  =  48. 

12  X 

3,  or  3  X 

12  =  36. 

.17  X  3,  or  3  X  17  =  51. 

13  X 

3,  or  3  X 

13  =  39. 

18  X  3,  or  3  X  18  =  54. 

14  X 

3,  or  3  X 

14  =  42. 

19  X  3,  or  3  X  19  =  57. 

15  X 

3,  or  3  X 

15  =  45. 

20  X  3,  or  3  X  20  ==  60. 

11  X 

4,  or  4  X 

11  =  44. 

16  X  4,  or  4  X  16  =  54 

12  X 

4,  or  4  X 

12  =  48. 

17  X  4,  or  4  X  17  =  68. 

13  X 

4,  or  4  X 

13  =  52. 

18  X  4,  or  4  X  18=:  72. 

14  X 

4,  or  4  X 

14  ==  5G. 

19  X  4,  or  4  X  19  =  76. 

15  X 

4,  or  4  X 

15  =  60. 

20  X  4,  or  4  X  20  =  80. 

100 

colburn's 

FIRST    PART. 

11  X 

5,  or  5 

X 

11  =  55. 

16  X  5,  or  5 

X  16  =  80. 

12  X 

5,  or  5 

X 

12  =  60. 

17  X  5,  or  5 

X  17  ==  85. 

13  X 

5,  or  5 

X 

13  =  65. 

18  X  5,  or  5 

X  18  =  90. 

14  X 

5,  or  5 

X 

14  =  70. 

19  X  5,  or  5 

X  19  =  95. 

15  X 

5,  or  5 

X 

15  =  75. 

20  X  5,  or  5 

X  20  =  100. 

11  X 

6,  or  6 

X 

11  =  66. 

16  X  6,  or  6 

X  16  =  96. 

12  X 

6,  or  6 

X 

12  =  72. 

17  X  6,  or  6 

X  17  =  102. 

13  X 

6,  or  6 

X 

13  =  78. 

18  X  6,  or  6 

X  18  =  108. 

14  X 

6,  or  6 

X 

14  =  84. 

19  X  6,  or  6 

X  19  =  114. 

15  X 

6,  or  6 

X 

15  =  90. 

20  X  6,  or  6 

X  20  =r  120. 

11  X 

7,  or  7  X 

11  =  77. 

16  X  7,  or  7 

X  16  =  112. 

12  X 

7,  or  7 

X 

12  =  84. 

17  X  7,  or  7 

X  17  =  119. 

13  X 

7,  or  7 

X 

13  =  91. 

18  X  7,  or  7 

X  18  =  126. 

14  X 

7,  or  7 

X 

14  =  98. 

19  X  7,  or  7 

X  19  =  133. 

15  X 

7,  or  7 

X 

15  =  105. 

20  X  7,  or  7  X  20  =  140. 

11  X 

8,  or  8 

X 

11  =  88. 

16  X  8,  or  8 

X  16  =  128. 

12  X 

8,  or  8 

X 

12  =  96. 

17  X  8,  or  8 

X  17  =  136. 

13  X 

8,  or  8 

X 

13  =  104. 

18  X  8,  or  8 

X  18  =  144. 

14  X 

8,  or  8 

X 

14  ==  112. 

19  X  8,  or  8 

X  19  =  152. 

15  X 

8,  or  8 

X 

15  =  120. 

20  X  8,  or  8 

X  20  =  160. 

11  X 

9,  or  9 

X 

11  ==  99. 

16  X  9,  or  9 

X  16  =  144. 

12  X 

9,  or  9 

X 

12  =  108. 

17  X  9,  or  9 

X  17  =  153. 

13  X 

9,  or  9 

X 

13  =  117. 

18  X  9,  or  9 

X  18  =  162. 

14  X 

9,  or  9 

X 

14  =  126. 

19  X  9,  or  9 

X  19  =  171. 

15  X 

9,  or  9 

X 

15  =  135. 

20  X  9,  or  9 

X  20  =  180. 

LESSON    FORTIETH. 


101 


11  X  10,  or  10  X  11  =  110. 

12  X  10,  or  10  X  12  =  120. 

13  X  10,  or  10  X  13  =  130. 

14  X  10,  or  10  X  1^  =■  140. 

15  X  10,  or  10  X  15  =  150. 


16  X  10,  or  10  X  10  ^  160. 

17  X  10,  or  lOx  17===:  170. 

18  X  10,  or  10  X  18  =  180. 

19  X  10,  or  10  X  19  =  190. 

20  X  10,  or  10  X  20  ==  200. 


LESSON  XL. 

Note.  —  Should  the  Teacher  deem  it  best,  the  class  may  omit  this 
nnd  the  next  three  Lessons,  till  after  some  of  the  first  Lessons  on 
Fractions  have  been  learned. 

A.  From  the  exercises  of  Lesson  XXXVII ,  B.  and  C,  we  may 
infer  that  4  times  30  =  40  times  8  ;  that  70  times  9  =  7  times 
90,  &c.,  &c.  In  like  manner,  40  times  27  =  4  times  270,  or,  4 
tens'  times  27  =  4  times  27  tens ;  80  times  436  =  8  times  4360, 
or,  8  tens'  times  436  =  8  times  436  tens,  &c.,  &c. 


1.  What  is  the  product  of  70  times  389  ? 

Solution.  —  70  times  389  =  7  times 
3890,  which  may  be  found  by  the  method 
explained  in  Lesson  XXXVIL,  G.  and 
H.  Thus  :  7  times  0  units  =  0  units.  7 
times  9  tens  =  63  tens  =  6  hundreds  and 
3  tens,  Ac,  &o. 


70 


2. 

20  times  64  ? 

3. 

80  times  29  ? 

4. 

40  times  36  ? 

5. 

60  times  94  ? 

6. 

90  times  37  ? 

7. 

20  times  93  ? 

8. 

30  times  84  ? 

9. 

90  times  72  ? 

26p30 

10.  30  times  979  ? 

11.  40  times  832  ? 

12.  70  times  697  ? 

13.  20  times  443  ? 

14.  60  times  927  ? 

15.  80  times  423  ? 

16.  50  times  975  . 

17.  30  times  476? 


9* 


lOi: 


COLBURN    S    FIRST     PART. 


B.  Since  24  ==  20  +  4,  24  times  any  number  must  equal  20 
times  that  number  plus  4  times  that  number.  Since  86  =  80  -}-  6, 
86  times  any  number  must  equal  80  times  that  number  plus  6 
times  that  number,  &c.,  &c. 

1.  What  is  the  product  of  29  times  863  ? 
S63 


2p 


Solution.  —  Since  29  =  20  +  9,  29  times 
863  must  equa^20  times  863,  plus  9  times  863. 
We  first  multiply  by  9,  and  then  by  20,  by  the 
methods  before  explained,  and  add  the  products 
together  as  in  the  written  work  at  the  left. 


7767 

^726 

250S7 


2.  38  times  481  ? 

3.  27  times  936  ? 

4.  68  times  427  ? 

5.  43  times  268  ? 

6.  31  times  492  ? 

7.  68  times  946  ? 

8.  79  times  368  ? 

9.  42  times  427  ? 

10.  54  times  329? 

11.  61  times  428? 


LESSON  XLI. 

A.  When  the  divisor  is  a  large  number,  it  is  often  convenient 
or  necessary  to  use  the  nearest  number  of  tens,  hundreds,  or  thou- 
sands, as  a  trial  divisor ,  to  determine  the  probable  quotient  figure. 


12. 

89  X  2796 ! 

13. 

38  X  9582  ? 

14. 

22  X  4858  ? 

15. 

56  X  9375  ? 

16. 

4^  X  2401  ? 

17. 

63  X  2485? 

18. 

81  X  3258? 

19. 

69  X  2846  ? 

20. 

44  X  8132  ? 

21. 

74  X  9123  ? 

LESSON    FORTY-FIRST. 


103 


Illustrations. — In  dividing  by  31,  32,  33,  or  34,  we  may  make  30 
or  3  the  trial  divisor.  In  dividing  by  36,  37,  38,  or  39,  we  may  make 
40  or  4  the  trial  divisor.  In  dividing  by  35,  we  may  make  either  30 
or  40  the  trial  divisor. 


B.     1.  What  is  the  quotient  of  178  -^  53? 

Solution. — "We  may  make  50  or  5  the  trial  divisor,  for  53  is  con- 
tained in  178  about  the  same  number  of  times  that  50  is  ;  or,  that  5  is 
contained  in  17,  which  is  3  times.  To  test  the  correctness  of  this  con- 
clusion, we  must  find  3  times  ^.  It  is  159,  which,  subtracted  from 
178,  leaves  19,  thus  showing  that  178  -^  53  =  3  with  19  remainder. 

The  work  would  be  written 
by  placing  the  divisor  at  the 
left  of  the  dividend,  the  quo- 
tient at  the  right,  and  the  pro- 
duct with  the  remainder  be- 
neath the  product. 


63 


W78  f3 
/5p  =  3x53 


/p ^e, 


Note.  —  The  Teacher  should  illustrate  and  explain  the  method  of 
proceeding  when  the  above  process  gives  a  trial  quotient  figure  either 
too  large  or  too  small.  [See  "Arithmetic  and  its  Applications,"  91st 
article,  and  solution  to  2d  example,  113th  page,  and  to  4th  example,. 
114th  page.]  .  -^ 


2.  96 

3.  127 

4.  228  • 
6.  683  • 

6.  281 

7.  469 

8.  356 

9.  429 


24? 
31? 
64? 
82? 
29? 
48? 
61? 
67? 


10. 

256- 

-  38? 

11. 

124- 

-  19? 

12. 

387- 

-45? 

13. 

621  - 

-84? 

14. 

438- 

-  62? 

45. 

279- 

-  94? 

16. 

349- 

-  82? 

17. 

624- 

-79? 

104 


COLBURN    S    FIRST    PART. 


C.     What  is  the  quotient  of  2856  -h  59  ? 


6pj 


2856  ^AS 
236 


Explanation.  —  69  is  eo 
near  60,  that  we  make  6  the 
trial  divisor.  Since  6  is  con- 
tained 4  times  in  28,  we  make 
4  the  first  figure  of  the  quo- 
tient, and  infer  that  59  is  con- 
tained 4  tens'  times  in  285 
tens.  Multiplying  59  by  4 
tens,  gives  236  tens.  Hence 
^  we  write  236  under  the  285, 
and  subtract  the  former  from  the  latter.  It  leaves  a  remainder  of  49 
or  49  tens,  490  units,  to  which,  adding  the  6  units,  gives  496  units. 
Since  6  is  contained  8  times  in  49,  we  make  8  the  next  figure  of  the 
quotient,  and  infer  that  59  is  contained  8  times  in  496.  Multiplying 
59  by  8,  gives  472;  hence  we  write  472  under  the  496,  and  subtract 
the  former  from  the  latter.  It  leaves  a  remainder  of  24.  Hence, 
2856  -^59-48  with  24  remainder. 

Proof. — 48  times  59,  plug  24,  equalf  2856. 


A96 
A72 

~YA=e^t 


em. 


2. 

843  H 

-  31? 

8. 

579  H 

-43! 

4. 

827- 

-15» 

5. 

1748  - 

-  42? 

6. 

3947- 

-49? 

7. 

8246- 

-  91? 

8. 

4217- 

-  88? 

9. 

8321  - 

r  94? 

10. 

6735- 

-  83? 

11. 

2317 -i 

-  88? 

12. 

7635- 

-  82? 

13. 

1749- 

-22? 

14. 

2175- 

r  25? 

15. 

4802- 

-  49? 

16. 

6237- 

-  74? 

17. 

4238- 

-  52? 

18. 

6947- 

r75? 

19. 

8286- 

-47? 

LESSON    FORTY-SECOND.  105 


LESSON  XLII. 

A.  1.  The  multiplier  and  multiplicand  are  called  factors  of 
their  product.     (See  Lesson  XXXII.,  B.) 

2.  The  FACTORS  of  any  number  are  the  numbers  which,  multi- 
plied together,  will  give  that  number  for  a  product. 

Illustrations.  —  6  and  3  are  factors  of  15,  because  15  =  5  X  3. 

Again,  6  and  2,  3  and  4,  or  3,  2,  and  2,  are  factors  of  12,  because  12 
=  6X2,  =  3X4  =  3X2X2. 

Again,  2  is  a  factor  of  4,  6,  8,  10,  &c.  3  is  a  fiictor  of  6,  12,  15, 
18,  &c, 

3.  From  the  above  illustrations,  we  see  that  the  factors  of  a 
number  are  divisors  of  it,  i.  <?.,  they  are  such  numbers  as  will 
divide  it  without  a  remainder. 

4.  A  prime  number  is  a  number  which  has  no  factors  except 
itself  and  1. 

Illustrations. — 1,  2,  3,  5,  7,  and  11,  are  each  prime  numbers. 

5.  A  composite  number  is  a  number  which  has  other  factors 
besides  itself  and  1. 

Illustrations.  —  4,  6,  8,  and  9,  are  each  composite  numbers,  for 
4«=2X  2,  6  =  3X2,  8  =  2X4  =  2X2X2,  and  9  =  3X3. 

6.  A  number  is  divided  into  factors,  when  any  factors  which 
will  produce  it  are  found. 

Illustrations. — In  '<  12  =*  6  X  2,"  12  is  divided  into  the  factors  6 
and  2,  but  in  "  12  -=  2  X  2  X  3,"  it  is  divided  into  the  factors  2,  2, 
and  3. 

7.  A  number  is  divided  into  its  prime  factors  when  it  is  divided 
into  factors  which  are  all  prime  numbers. 

Illustrations.— 18  =-2X3X3.  80  =  2  X  3  X  5. 


8.  The  product  of  a  number  taken  any  number  of  times  as  a 
factor,  is  called  a  power  of  that  number. 

Illustrations.  —  8,  which  is  the  product  of  2  X  2  X  2,  i.  e.,  of  2 
taken  3  times  as  a  factor  is  the  third  power  of  two;  25,  which  is  the 
product  of  5  X  5,  t.  e.,  of  5  taken  2  times  as  a  factor  is  the  second  power 
of  five. 

9.  We  may  indicate  the  power  of  a  number  by  writing  a  small 
figure,  called  an  exponent,  above  it  and  a  little  to  the  right. 

Illustrations.— 3  '  =  3  X  3  X  3,  or  3  to  the  third  power.  2 »  =  2 
X  2  X-  2  X  2  X  2  ,  or  2  to  the  fifth  power. 

10.  The  second  power  of  a  number  is  sometimes  called  its 
square,  and  the  third  power  its  cube.  Thus,  8,  or  2 ',  is  the  cube 
of  2 ;  25,  or  5*,  is  the  square  of  5. 

B.  Write  the  prime  factors  of  the  numbers  from  1  to  100,  as 
in  the  following  model : — 

1  =  Prime.  6  =  2x3. 

2  =  Prime.  7  =  Prime. 

3  =  Prime.  8  =  2x2x2  =  2' 

4  =  2x2  =  2'.  9  =  3  X  3  =  3^ 

5  =  Prime.  10  =  2  X  5. 

C.  A  factor  is  common  to  two  or  more  numbers  when  it  is  a 
factor  of  each  of  them. 

Illustrations. — 2  is  a  common  factor  of  4,  ^  and  14,  for  it  is  a 
factor  of  each. 

1.     What  prime  factors  are  common  to  24,  36, 
and  48  ? 

Dividing  each  number  into  its  prime  factors,  gives  — 
Solution.— 24  =^=2x2x2x3=2^X3. 
36  =  2  X  2  X  3  X  3  =  2'  X  3*. 
48  =  2x2x2x2x3  =  2*  X  3. 


LESSON    FORTYrSECOND.  107 


By  inspecting  these,  we  see  that  2'  and  3  are  factors  of  each 
number,  and  that  there  is  no  other  common  factor.  Hence  2,  2, 
and  3,  or  2  **  and  3  are  the  prime  factors  required. 

What  prime  factors  are  common  — 


3. 

To  12  and  18  ? 

10. 

To  6,  8,  and  10? 

4. 

To  15  and  25? 

11. 

To  12,  18,  and  30  ? 

5. 

Toll  and  20? 

12. 

To  20,  30,  and  50? 

6. 

To  30  and  40  ? 

13. 

To  42,  56,  and  84  ? 

7. 

To  36  and  54  ? 

14. 

To  63,  81,  and  99? 

8. 

To  7  and  9  ? 

15. 

To  8,  9,  and  25  ? 

9. 

To  39  and  54  ? 

Ifc 

To  7,  49,  and  84  ? 

D.  A  Common  Divisoii  of  two  or  more  numbers  is  any  number 
which  will  exactly  divide  each  of  them. 

Illustration. — 4  is  a  common  divisor  of  4,  8, 12,  and  32. 

The  Greatest  Common  Divisor  of  two  or  more  numbers  is  the 
largest  number  which  is  a  divisor  of  each  of  them.  It  is  also  the 
product  of  all  their  common  prime  factors. 

1.  What  is  the  greatest  common  divisor  of  24, 
36,  and  60  ? 

Solution.  —  The  greatest  common  divisor  of  24,  36,  and  60,  is  the 
product  of  all  the  pfime  factors  common  to  these  numbers. 

24  =  2x2x2x3  =  2^x3. 

36  =  2  X  2  X  3  X  3  =  22  X  32. 

60  =  2x2x3x5  =  2^x3x5. 

We  see  that  the  only  common  prime  factors  are  2,  2,  and  3. 
Hence  2  X  2  X  3,  or  12,  must  be  the  greatest  common  divisor 
required. 


108  colburn's  first   part. 


What  is  the  greatest  common  divisor  — 

2.  Of  30  and  42?  9.  Of  4,  6,  and  12? 

3.  Of  4  and  12?  10.  Of  3,  9,  and  15? 

4.  Of  8  and  20  ?  11.  Of  18,  27,  and  45  ? 
6.  Of  35  and  49  ?  12.  Of  14,  28,  and  56  ? 

6.  Of  63  and  72  ?  13.  Of  30,  48,  and  54  ? 

7.  Of  42  and  63  ?  14.  Of  24,  60,  and  84  ? 

8.  Of  72  and  96  ?  15.  Of  45,  75,  and  90  ? 


LESSON  XLIII. 

A.  The  product  of  any  numbers  is  sometimes  called  their  mul- 
tiple. Thus,  12  is  a  multiple  of  1,  of  2,  of  3,  of  4,  of  6,  and  of 
12,  for  it  equals  1  X  12,  or  2  x  6,  or  3  x  4. 

Hence,  any  number  is  a  multiple  of  all  its  factors  and  divisors, 
and  a  factor  of  all  its  multiples. 

Every  multiple  of  a  number  must  contain  all  the  prime  factors 
of  that  number. 

1.  What  prime  factors  must  every  multiple  of 
18  contain  ? 

Solution. — Since  18  =  2  X  3',  every  multiple  of  18  must  contain 
the  factors  2  and  '6  \  or  2,  8,  and  3. 

What  prime  factors  must  be  contained  in  every 
multiple  — 

2.  Of  12?  5.     Of  33?  8.     Of  48? 

3.  Of   9?  6.     Of 28?  9.     Of 60? 

4.  Of  21?  7.     Of  75?  10.     Of  36? 

B.  A  Common  Multiple  of  several  numbers  is  a  number  which 
is  a  multiple  of  all  of  them. 


LESSON    FORTY-THIRD.  109 

The  Least  Common  Multiple  of  several  numbers  is  the  least 
number  which  is  a  multiple  of  all  of  them,  and  is  therefore  the 
smallest  number  which  contains  all  the  prime  factors  of  each  of 
them. 

1.  What  is  the  least  common  multiple  of  36  and 

48? 

Solution. — Tho  least  common  multiple  of  36  and  48  is  the  smallest 
number  which  contains  all  their  prime  factors. 

36  ==  2  X  2  X  3  X  3  ==  2^  X  3'. 
48  =  2x2x2x2x3  =  2*  3. 

Solution. — "We  must  have  48  or  its  factors,  which  are  2  *  X  3.  We 
must  also  have  the  factors  of  36,  which  are  2*  X  3  2,  but  as  we  have 
already  taken  2  '  X  3,  we  have  only  to  introduce  the  remaining  factor 
3,  which  gives  48  X  3,  or  2  *  X  3  *  =  144,  as  the  L.  C.  M  required. 

2.  What  is  the  least  common  multiple  of  9,  24, 
and  30? 

Solution. — The  least  common  multiple  of  these  numbers  is  the  least 
number  which  contains  all  the  prime  factors  of  each  of  them. 

9  =  3X3. 

24  =  2X2X2X3. 

30  =  2  X  3  X  5. 

We  must  have  30  or  its  factors,  which  are  2  X  3  X  5.  We  must  also 
have  the  factors  of  24,  which  are  2  X  2  X  2  X  3  j  but  as  we  have 
already  taken  2  X  3,  we  have  only  to  introduce  the  remaining  factors 
2x2,  which  will  give  2  X  3  X  5  X  2  X  2,  or  30  X  2  X  2.  We  must 
have  the  factors  of  9,  which  are  3  and  3,  but  as  we  have  already  taken 
one  S,  we  have  only  to  introduce  another  3,  which  gives  as  the  L.  C.  M. 
required,  2X  3  X5x2x2x  3,  or  30  X2X2X3  =  360. 

What  is  the  least  common  multiple  — 

8.  Of  8  and  12  ?  6.  Of  2,  4,  and  6  ? 

4.  Of  6  and  9?  6.  Of  9,  12,  and  18? 


110 


colburn's   first   part. 


10. 


7.  Of    4  and  12  ? 

8.  Of    7  and    8  ? 

9.  Ofl2andl5? 
Of  16  and  20? 


11.  Of  10,  25,  and  30? 

12.  Of    4,    6,  and  12  ? 

13.  Of  3,  4,5,  and    6? 

14.  Of  8,  10,  12,  and  20? 


XoTE.— If  the  class  have  time  for  it,  the  Teacher  will  do  well  to 
give  them  the  method  explained  in  Arithmetic  and  its  ArPLiCATiONs, 
page  154. 


LESSON  XLIV. 


Oral  Exercise. — Exhibit  any  convenient  thing,  as  an  apple,  to 
the  class,  and,  cutting  it  into  two  equal  parts,  ask,  "What  have 
I  done  to  the  apple?"  Ans. — "You  have  cut  it."  "Into  how 
many  parts  have  I  cut  it?"  A7is. — "  Into  two  parts."  "  How  do 
the  parts  compare  in  size  ?"  Ans. — "  They  are  equal."  "  Then 
how  have  I  divided  the  apple  ?"  Ajis.  —  "  You  have  divided  it 
into  two  equal  parts."  "When  anything  is  divided  into  two  equal 
parts,  the  parts    are  called  halves  of    the  thing.      What,  then, 


LESSON    FORTY-FOURTH.  Ill 


will  you  call  these  parts  of  an  apple?"  Ans.  —  "  Halves  of  an 
apple."  "  What  will  you  call  one  part  ?"  Ans.  —  "  One-half  of 
an  apple."  "What  will  you  call  both  parts?"  Am.  —  "Two 
halves  of  an  apple."  *'  What  do  both  together  equal  ?"  Ans.  — 
'•  \  whole  apple."  *'  Then  how  many  halves  of  an  apple  equal  a 
whole  one  ?"     Ans. — "  Two  halves  of  an  apple." 

Continue  and  extend  these  illustrations  by  exercises  similar  in 
character  to  these  suggested  below :  — 

"  How  shall  I  divide  this  apple  (showing  another)  into  halves?" 
Dividing  it,  ask,  "  How  many  halves  have  I  from  it  ?  How  many 
halves  did  I  have  from  the  first  apple  ?  How  many  halves  are 
there  in  all  ?  Then  two  halves  and  two  halves  are  how  many 
halves  ?  If  I  should  give  away  one-half,  how  many  halves  should 
I  have  left  ?  Then  one-half  from  four  halves  leaves  how  many 
halves  ?" 

Vary  these  exercises,  dividing  apples,  strings,  pieces  of  paper, 
lines,  &c.,  till  the  class  understand  fully  the  value  of  halves, 
thirds,  &c.,  and  see'  clearly  that  they  can  be  added,  subtracted, 
multiplied,  and  divided  as  whole  numbers  are.  Such  a  course 
will  save  much  hard  labor  afterwards,  both  to  Teacher  and  pupil. 

A.  Explanations. — 1.  Such  parts  as  are  obtained  by  dividing  any- 
thing or  any  number  into  two  equal  parts,  are  called  halves  of  that 
thing  or  number.  One  such  part  is  called  one  half;  two  such  parts 
are  called  two  halves  ;  three  such  parts  are  called  three  halves, 
&c.,  <fcc. 

2.  Such  parts  as  are  obtained  by  dividing  anything  or  number  into 
3  equal  parts,  are  called  thirds  of  the  thing  or  number.  One  such 
part  is  called  one  third,  two  such  parts  are  called  two  thirds,  three 
such  parts  are  called  three  thirds,  four  such  parts,  four  thirds, 
<tc.,  &c. 

3d,  In  like  manner  such  parts  as  are  obtained  by  dividing  anything 
or  number  into  four  equal  parts  are  called  fourths  of  the  thing  or 
number;  such  as  are  obtained  by  dividing  it  into  five  equal  parts,  are 


112  colburn's  first  part. 

called   fifths;  into   six,  are  called  sixths;   into  seven,  are  called 

SEVENTHS,  Ac,  <tc. 

4.  Parts  like  these  are  called  fractional  parts. 


B.     1.  What  are  sevenths? 

Ans. — Sevenths  of  any  thing  or  number  are  such  fractional  parts 
as  would  be  obtained  by  dividing  it  into  seven  equal  parts. 
In  the  same  way  explain  each  of  the  following : — 


2. 

Fifths  ? 

6.     Twelfths  ? 

8. 

Fourths  ? 

3. 

Thirds? 

6.     Halves  ? 

9. 

Twentieths  ? 

4. 

Ninths  ? 

7.     Sixths? 

10. 

Tenths? 

C.  1.  Instead  of  the  word  sixihsy  we  may  write  the  figure  6 
with  a  line  above  it,  thus  :  r.  Instead  of  the  word  halves^  we  may 
write  2,  &c.,  &c. 

Hence  f  tj  means  tenths,  which  are  fractional  parts  of  such  kinds 
as  are  obtained  by  dividing  a  unit  into  10  equal  parts. 

■5  means  eighths,  which  are,  &c. 
z  means  thirds,  which  are,  &c. 
i\  means  twenty-firsts,  which  are,  &o. 
s  means  fifths,  which  are,  &c. 

The  number  which  thus  shows  into  how  many  parts  a  unit  is 
divided,  is  called  a  denominator,  because  it  gives  a  name  or 
denomination  to  the  parts.  To  indicate  that  a  number  is  a  denomi- 
nator, draw  a  little  line  over  it. 

D.  1.  How  many  sixths  does  it  take  to  equal  the 
whole  of  anything  ? 

Ans. — Six,  because  sixths  are  such  parts  as  are  obtained  by 
dividing  a  unit  into  six  equal  parts. 


LESSON    FOKTY-FOURTH.  113 


2.  How  many  ninths  does  it  take  to  equal  the 
whole  of  anything  ? 

3.  How  many  twelfths  ?  6.     How  many  -ju  ? 

4.  How  many  halves  ?  7.     How  many  t?  ? 

5.  How  many  thirds  ?  8.     How  many  ?  ? 

E.  In  order  to  show  how  many  fractional  parts  are  taken  or 
considered,  a  figure  is  written  above  the  denominator. 

Illustrations.  —  To  express  three  fourths,  the  figure  3  may  be 
written  above  the  denominator  4,  thus  ;  |. 

Five  sixths  may  be  written  ^ 

Eight  fifteenths  may  be  written  ^j. 

In  like  manner,  J  =  4  ninths ;  { |  =  13  seventeenths ;  and 
f  f  =  29  thirty-firsts. 

The  figure  which  thus  enumerates  or  numbers  the  parts,  is  called 
the  NUMERATOR,  and  shows  how  many  parts  are  taken  or  consi- 
dered. The  numerator  is  written  above  the  denominator,  and 
separated  from  it  by  a  line. 

F.  Such   expressions    as    "two-thirds,"    "three-elevenths," 

**T7»"  "|»"  ^^-J  ^^•»  ^^®  called  FRACTIONS. 


1.  What  does  the  fraction  |  express  ? 
Ans. — The  fraction  three-fourths  expresses  the  value  of  3  such 
parts  as  are  obtained  by  dividing  a  unit  into  4  equal  parts. 

In  the  same  manner  explain  each  of  the  following  fractions :  — 

2.     f.  4.     |.  6.     ^.  8.    if. 

G.     Fractions  may  be  explained  after  the  following  model :  — 
The  fraction  three  fourths  expresses  the  value  of  three  equal 
parts  of  such  kind  that  four  of  them  would  equal  a  unit. 


114  colburn's   first   part. 


1.  Explain  each  of  the  fractions  under  F.  according  to  the  last 
model. 

2.  What  is  the  numerator  and  what  the  denominator  of  each  ? 

H.     A  mixed  number  is  a  whole  number  and  a  fraction. 
Illustrations. — 4§,  which  is  read  "four  and  two-thirds." 

Read  each  of  the  following :  — 

1.    5  J.  2.    5?.  3.    4f.  4.    28  J. 

I.     These  illustrations  suggest  the  following  definitions : — 

1.  Fractional  parts  of  any  thing^  quantity  or  number^  are  such 
parts  as  are  obtained  by  dividing  it  into  equal  partB.     Or  — 

2.  Fractional  parts  of  any  thing,  qttantify  or  nmnber,  are  equal 
partit  of  such  kind  that  a  given  number  of  them-  will  equal  that  thing, 
quantity^  or  number, 

3.  A  FRACTION  expresses  the  value  of  one  or  more  such  parts 
as  are  obtained  by  dividing  a  unit  into  equal  parts.     Or — 

4.  A  FRACTION  expresses  the  value  of  one  or  more  such  equal 
parts  that  a  given  number  of  them  will  equal  a  unit. 

5.  The  number  which  shows  into  how  many  parts  the  thing  is 
divided,  or  how  many  of  the  parts  are  equal  to  a  unit,  is  called 
the  DENOMINATOR  of  thc  fractiou. 

6.  The  number  which  shows  how  many  parts  are  taken  or  consi- 
dered, is  called  the  numerator  of  the  fraction. 

7.  The  denominator  is  so  called  Jbecause  it  gives  the  name  or 
denomination  to  the  parts. 

8.  The  numerator  is  so  called  because  it  enumerates  the  parts 
taken  or  considered. 

.]    Write  ench  of  the  following  in  fijruros: — 


LESSON    FORTY-FIFTH.  115 


1.  Two- thirds.  5.  Four  and  seven- tenths. 

2.  Eight-ninths.  6.  Ten  and  four-fifths. 

3.  Thirteen-nineteenths.  7.  Twelve  and  eleven-twelfths. 

4.  Six  twenty-firsts.  8.  Six  and  two-thirds. 


LESSON  XLV. 

A.  Fractions  may  arise  from  division  as  in  the  following 
examples :  — 

29  =  *  times  6  V 

Solution.  —  29  ==  4  times  6,  with  5  remaining,  or  it  equals  4| 
tirnes  6. 

Note. — The^  first  part  of  the  above  solution  should  be  omitted  as 
soon  as  the  pupil  is  prepared  to  give  the  final  answer  without  it.  The 
entire  dividend  is  here  divided,  and  ihQ  fraction  Jive  sixths  is  apart  of 
the  quotient,  and  not,  like  the  remainder  5,  a  part  of  the  dividend. 
Hence  it  is  wrong  to  say  "  29  =  4  times  6,  with  |.  remaining."  These 
distinctions  are  important,  and  should  be  observed  in  the  solutions. 
See  Lesson  XXXVL,  Note  under  A. 

Perform  by  the  above  solution  the  examples  under  Letter  A.  and 
B.,  Lesson  XXXVL 

B.  1.  IIo-w  many  quarts  of  vinegar  at  6  cents 
per  quart,  can  be  bought  for  53  cents  ? 

Solution. — Since  1  quart  of  vinegar  can  be  bought  for  6  cents,  as 
many  quarts  can  be  bought  for  63  cents  as  there  are  times  6  in  53, 
which  are  8|  times.  Hence,  8|  quarts  of  vinegar  at  6  cents  per  quart 
can  be  bought  for  63  cents. 

2.  How  many  pounds  of  sugar  at  8  cents  per  pound,  can  be 
bought  for  68  cents? 

8.  How  many  yards  of  cloth  at  $4  per  yard,  can  be  bought  for 
$31? 


116  COL  burn's   first   part. 


4.  How  many  bags,  each  containing  3  bushels,  can  be  filled 
from  29  bushels  of  grain  ? 

5.  How  many  hours  will  it  take  a  horse  to  trot  33  miles,  if  he 
trots  7  miles  per  hour  ? 

6.  How  many  weeks  will  it  take  a  man  to  earn  $78,  if  he  earn 
$9  per  week? 

7.  How  many  hours  will  it  take  a  ship  to  sail  63  miles,  if  she 
sail  8  miles  per  hour  ? 

8.  How  many  months  wUl  it  take  a  man,  -who  earns  $12  per 
month,  to  earn  $105? 

9.  A  man  put  9  bu.  3  pk.  of  grain  into  bags,  each  holding  1 
bu.  3  pk.     How  many  bags  could  he  fill? 

10.  If  a  man  can  earn  enough  in  one  day  to  buy  1  gal.  2  qts. 
of  oil,  how  many  days  will  it  take  him  to  earn  enough  to  buy  13 
gal.  1  qt.  ? 


LESSON  XLVI. 

A.  Fractions  may  be  added,  subtracted,  multiplied,  and  divided 
as  whole  numbers  are.     Thus : — 

2 4  =  6  Just  as  2  days  +  4  days  =  6  days. 

1 J  =  I,  just  as  5  qts.  —  3  quarts  =  2  quarts. 

9  times  |  ==  |^,  just  as  9  times  3  pecks  =  27  pecks. 

S>  are  contained  4  times  in  ||,  just  as  6  lb.  are  contained  4 
times  in  24  lb. 

B.     1.  7  =  *  fourths  ? 

Solution.  —  Since  1  =  4  fourths,  7  must  equal  7  times  4  fourths, 
which  are  28  fourths.     Therefore,  7  =<  ^s. 


LESSON    FORTY 

-SIXTH.                   117 

2. 

81 

=  *  fifths  ? 

Solution.  - 
are  40  fifths, 

-  Since  1  =-  5  fifths, 
and  4  fifths  added  an 

8  must  equal 
)  44  fifths. 

8  times  5  fifths,  which 
Hence  8|  =Y- 

NOTE.- 

-Compare  these  solutions  with  those  of  E.,  Lesson  XXX 

3.    9 

^  tenths  ? 

7. 

4-^ 

=  *  seventeenths? 

4.    5 

:  *  thirds  ? 

8. 

2A 

=  *  nineteenths  ? 

5.    8 

^  nineteenths  ? 

9. 

7}?- 

=  *  seventeenths  ? 

6.      4J=: 

^  fourths  ? 

10. 

6/r 

=  ^  elevenths? 

c. 

1. 

5^8  =:  Hi  ones  ? 

SOLUTION.- 

as  there  are  t 

—Since  9  ninths  =<  1,  58  ninths  must  equal  as  many  ones 
imes  9  in  58,  which  are  6J  times.     Hence,  -''-^  =  6|. 

NOTE.- 

-Compare  this  solution  with  th 

)se  of  F.,  Lesson  XXX. 

2. 

V 

=  ^  ones? 

7. 

%«    =  *  ones  ? 

3. 

27 
5 

=  ^  ones  ? 

8. 

f r    =  -3^  ones? 

4. 

V 

=  *  ones? 

9. 

Y/  =  *  ones? 

5. 

33 

=  ^-  ones? 

10. 

V    =  -5^  ones  ? 

6. 

V 

=  *  ones  ? 

11. 

II    =  -jf  ones? 

D. 

1. 

What  is  the  sum  of 

A  +  t\? 

1st  Solution.— -j9-  -f  JL  =  js 

=  1tV 

2d  Solution.  —  Observing  that 
,-"t  +  t\  =  tV  +  T>  +  T'f  = 

/t  = 

ll,  or  1,  we  may  have 

Note. — The  second  forms  of  solution  to  the  problems  under  D.  and    J 
G.,  and  the  third  form  under  E.,  will  often  be  found  much  easier  than 
the  first. 

,     - ....                                                                              1 

118 

COLBURN 

'S    FIRST    PART. 

2.    5  +  1? 

5.     ?  + 

f+4+i? 

3.    t'+}J? 

6.    1  + 

f+l  +  l? 

4.    I'l+iJt 

7.    A  + 

A+iV+ii? 

E. 

1.      4+3  +  i? 

Then  4  +  3  i  ? 

2.      8  +  7  +  1  ? 

Then  8  +  7i? 

3.    12+21  +  f? 

Then  12  +  21, 

J» 

4. 

What  is  the  sum 

0f6|  +  8|? 

1st  Solution.— 6|  and  8  an 

1  14  J,  and  '  are  14J  =  15| 

2d  Solution.  — 6  4-8  =  14;  j  +  |  =  J  = 
14  =  15  f . 

1|,  which,  added  to 

3d  Solution.  —  Observing 

that  6J  +  J  =  7,  we  have  6j  +  8^  = 

6|  + 

•  +  8|  =  r  +  8|  = 

16|. 

5. 

9? +6?? 

9-    Si%+'!i%+&tV 

6. 

3I  +  4J? 

10.    5|  +  7| 

+  4J+5I? 

7. 

4A+3/tT 

11.    8J  +  2| 

+  7J+3f? 

8. 

Hi  +  ^V 

12.    5J  +  6| 

+  55  +  45? 

F. 

1.  6?-3i? 

Solution.— 6J  —  3  ==  3  J  ; 

3J  — |=:3|.    Hence,  67— 3|  =  S|. 

2. 

8f  —2\1 

5. 

16J   -9|? 

3. 

14|  -7|? 

6. 

1513  _4j\? 

4. 

Wt-4t'5? 

7. 

38ii  -  292J  ? 

G. 

1.  What  is  the  value  of  1  — 

t\? 

SoLUTios.— 1  =•  if,  and  j| 

-TV  =  tV 

LESSON    FORTY-SIXTH 

119 

2.  What  is  the  value  of  8 

-^u 

? 

Solution.  ~  8  -  -^^  «  7|J  -  /^  = 

=  Ui 

3.    i  —  i? 

6. 

3-1? 

4.    1-if? 

7. 

S-fVf 

5.     1-ii? 

8. 

9-1? 

9.  8^^,-13? 

1st  Solution.— 8-''^  =  7  +  l^"^^  — 

i?  = 

m- 

i5  =  HI. 

2d  Solution.  -8-^^  — 13  —  sJ^^  ~ 

•r\- 

^%  = 

^-i%=^m- 

10.  23y5^^16A? 

1st  Solution.~23-j:^^  —  16  =  7^^ 

=  6|B.  6|| 

-  r%  =  «tV 

2d  Solution.  —  23^5^  —  10  =  7/^ 

;  Vj 

-t\ 

=  h'i  -  A  - 

11.       9i    -7|? 

15. 

23A 

-13/,? 

12.       4A-/,? 

IG. 

8/f 

-3ii? 

13.     16xV-5f2? 

17. 

64J 

+  4J  -  gp 

14.    43f   —  17|? 

18. 

23| 

+  17|  -  8J  ? 

H.     1.  George  had  a  very  large  apple. 

He  gave  William  J  of 

it,  Joseph  1  of  it,  and  ate  the  rest. 

What 

part  of  it  did  he  eat  ? 

2.  Edward  earned  |  of  a  dollar  by  picking  blackberries,  |  of  a  || 

dollar  by  picking  strawberries,  and 

J  of  a 

dollar 

by  picking  blue- 

berries.     How  much  did  he  earn  in 

all? 

3.  Who  can  tell  whether  the  sum  of  S 
.greater  or  less  than  24|,  and  how  much  ? 

1  +  5 

1  +  8|  +  7|  is 

4.  From  a  lot  containing  8^  acres,  there  were  5|  acres  sold. 

How  many  acres  were  left  ? 

, 

120  COLBURN*S    FIRST    PART. 


5.  Mr.  Stone  gave  -f-^  of  his  money  for  a  lot  of  land,  and  ^  for 
a  horse.     What  part  of  it  had  he  left  ? 

6.  Isaac  caught  three  nice  trout.  The  first  weighed  3-j-'^^  lb., 
the  second  -weighed  2j|  lb.,  and  the  third  weighed  lj|  lb.  How 
much  did  they  all  weigh  ? 

7.  Mr.  Davis  owns  -^^^  of  a  vessel,  Mr.  Mason  owns  ^®^,  and 
IsIt.  Allen  owns  the  rest.  What  part  of  the  vessel  does  Mr.  Allen 
own? 

8.  Julia's  father  gave  her  ||  of  a  dollar,  her  mother  gave  her 
]  J,  her  brother  gave  her  ^|,  and  her  uncle  gave  her  enough  to 
make  up  2  dollars.     How  much  did  her  uncle  give  her  ? 

9.  Farmer  Brown  had  VI-^^  tons  of  hay  in  his  barn,  and  15^'^^ 
tons  in  his  stacks.  How  many  tons  had  he  in  both  ?  He  moved 
o\^  tons  from  his  stacks  into  his  barn.  How  many  tons  were 
then  in  his  bam  ?     How  many  in  his  stacks  ? 

10.  A  man  paid  11 J  dollars  for  a  coat,  3J  dollars  for  a  pair  of 
pants,  and  2|  dollars  for  a  vest,  giving  in  payment  a  twenty-doUar 
bill.     How  much  ought  he  to  receive  back  ? 


LESSON  XLVII. 
A.     1.  9  times  JJ  ? 

Solution. — 9  times  ^  =  Y>  which,  since  ^  =  1,  must  equal  as 
many  ones  as  there  are  times  7  in  64,  which  are  7|  times.  Hence,  9 
times  ^  «x  75, 

Abbreviated  Solution. — 9  times  ^  =.  *j4  js-  75. 

2.     4  times  /^  ?  6.     9  times  |  ? 


7.     4  times  /^  ? 
? 

5.     7  times  jl  ?  9.     6  times  11  ? 


4.     4  times  «?  8.     5  times  f? 


LESSON    FORTY-SEVENTH.               121 

10. 

6  times  7|  ? 

SoLUTipN.— 6  times  7  =  42,  and  6  times  ^  =  l^  =.  33,Tvhich,  added 
to  42,  gives  453.     Hence,  6  times  7|  ==  45|. 

11. 

9  times  8f?                          '          15.     6  times  8^9^? 

12. 

8  times  93?                                    16.     5  times  9f? 

13. 

7  times  4|?                                    17.     4  times  8jJ  ? 

14. 

4  times  6}f  ?                                  18.     6  times  14/g  ? 

B. 

1.  12|  =  *  times  2i  ? 

Solution. — 12|  =  ^^  ',  2|-  =  | ;  and  ^j  contains  ^'  as  many  times 
as  51  contains  9,  which  are  5|  times.     Hence,  123.  =  5|  times  2^, 

2.  81-^3? 

Solution.— 8|  «  %«  ;  3  =.  f ;  and  2^«  -^  |  =  26  -^  9  ==2|.  Hence 
8|  -r-  3  =  23. 

3. 

7i  =  *  times  1}?                               8.     TJ —  IJ? 

4. 

4|  =  ^  times  If?                               9.     8f~2J? 

6. 

84.  ==  *  times  If  ?                             10.     9  J  H-  3  J  ? 

6. 

9§==  ^  times  2|?                              11.     Gf-f-la? 

7. 

58  =  *  times  J?                                12.     8|-MJ? 

C.     ] 
yard 

I    How  much  will  7  yards  of  cloth  cost  at  9^  dollars  per 

2.  A 

$7  per 

man  gave  6  cords  of  wood  at  $42  per  cord  for  raisins  at 
cask.     How  many  casks  did  he  buy  ? 

3.  How  many  yards  of  cloth,  at  $3  per  yard,  can  be  bought  for 
8  bbls.  of  cider  at  $3|  per  barrel  ? 

10 


122  colburn's   first   part. 


4.  I  gave  9  firkins  of  butter  at  $4|  per  firkin  for  flour  at  $7 
per  barrel.     How  many  baiTels  did  1  buy  ? 

6.  How  many  shade  trees,  worth  |  of  a  dollar  a-piece,  can  be 
bought  for  51  dollars? 

6.  How  many  skeins  of  silk,  worth  -J  of  a  dime  per  skein,  can 
be  bought  for  6|  dimes  ? 

7.  How  many  baskets,  each  containing  ^  of  a  bushel,  can  be 
filled  from  8^  bushels  of  peaches  ? 

8.  How  many  boxes,  each  holding  J  of  a  quart,  can  be  filled 
from  7 1  quarts  of  blackberries  ? 

9.  A  man  paid  6J  dollars  for  grain  at  J  of  a  dollar  per  bushel. 
How  many  bushels  did  he  buy?  He  put  the  grain  into  bags 
each  holding  IJ  bushels.     How  many  bags  did  he  fill? 

10.  Ralph  paid  5|  dimes  for  parched  corn  at  |  of  a  dime  per 
quart.  How  many  quarts  did  he  buy  ?  After  giving  away  1  of 
a  quart,  he  put  the  rest  into  paper  bags  each  holding  |  of  a  quart. 
How  many  bags  did  it  take  ? 

11.  A  farmer  exchanged  5  barrels  of  apples  at  1§  dollars  per 
barrel,  for  oil  at  lij  dollars  per  gallon.  How  many  gallons  of  oil 
did  he  get? 

12.  How  many  pounds  of  tea,  worth  ^  of  a  dollar  per  pound, 
should  be  given  for  three  books  worth  2^  dollars  a-piece  ? 

13.^  A  man  who  had  lOJ  bushels  of  potatoes,  used  2 J  bushels, 
and  then  sold  the  rest  at  J  of  a  dollar  per  bushel,  receiving  in 
payment  cloth  at  J  of  a  dollar  per  yard.  IIow  many  yards  of 
cloth  did  he  receive  ? 

14.  I  bought  8  barrels  of  flour  at  $7^  per  barrel,  and  gave  in 
payment  12  cords  of  wood  at  $4§  per  cord,  and  the  rest  in  apples 
at  4  of  a  dollar  per  bushel.  How  many  bushels  of  apples  did  I 
sive? 


LESSON    FORTY-EIGHTH.  123 


LESSON  XLVIII. 

A.  1.  What  part  of  5  is  1  ? 
^7w. — 1  is  ^  of  6,  because  5  times  1  =  5. 

What  part  — 

2.     Of  7  is  1?  6.  Of  10  is  1? 

8.     Of  2  is  1?  6.  Of  Sis  1? 

4.     Of  9  is  1?  7.  Of  Sis  1? 

B.  1.  What  part  of  8  is  3  ? 

Solution. — Since  1  =  J  of  8,  3  must  equal  f  of  8. 

What  part  — 

2.  Of  12  is  7  ?  6.  Of  9  is  10  ? 

3.  Of  9  is  4?  7.  Of  13  is  11? 

4.  Of7is2?  8.  Of8is5? 

5.  Of  10  is  9?  9.  Of  5  is  8? 

C.  1.  What  part  of  9  quarts  is  4  quarts  ? 

Solution, — i  quarts  is  the  same  part  of  9  quarts  that  4  is  of  9,  which 

What  part  — 

2.  Of  8  yd.  is  3  yd.  ?  6.  Of  $5  is  $3? 

3.  Of  11  lb.  is  7  lb.  ?  6.  Of  £12  is  £5? 

4.  Of  7  lb.  is  11  lb.?  7.  Of  £5  is  £12? 

8.  "V^Tiat  part  of  the  cost  of  7  yd.  is  the  cost  of  4  yd.  ? 

9.  What  part  of  the  cost  of  3  acres  is  the  cost  of  4  acres  ? 


124  colburn's  first   part. 


10.  What  part  of  the  cost  of  10  bushels  is  the  cost  of  9  bushels? 

11.  When  flour  is  9  dollars  per  barrel,  what  part  of  a  barrel  can 
be  bought  for  1  dollar  ?     For  5  dollars  ? 

12.  Mr.  Edwards  and  Mr.  Boyden  bought  a  cask  of  oil,  contain- 
ing 8  gallons,  which  they  so  divided  that  Mr.  Edwards  had  3 
gallons,  and  Mr.  Boyden  had  6.  What  part  of  the  cost  should 
each  pay  ? 

13.  Mr.  Avery,  Mr.  Leavens,  and  Mr.  Congdon  together  bought 
17  bushels  of  corn,  which  they  so  divided  that  Mr.  Avery  took  4 
bushels,  Mr.  Leavens  took  6  bushels,  and  Mr.  Congdon  took  the 
remainder.     What  part  of  the  cost  should  each  pay  ? 


LESSON  XLIX. 
A.     1.  What  is  i  of  63? 

Ans. — 1  of  63  is  7,  because  9  times  7  =  t)3. 

2.  1-  of  24  ?  4.     J-  of  49  ?  6.     |  of  64  ? 

3.  ^of25?  6.     4  of  42?  7.     ^jt  of  81  ? 

8.  What  is  I  of  72  ? 

1st  Solution.— i  of  72  =  3  times  t  of  72 ;  J  of  72  is  9,  and  3  times 
9  =  27.     Hence  |  of  72  =  27. 

2d  Solution. — J  of  72  =  9,  and  f  of  72  must  equal  3  times  9,  which 
are  27.     Hence,  §  of  72  =  27. 

9.     fof40?  12.     7  of  56?  15,     J  of  36? 

10.  fofl8?  13.     3?^  of  80?  16.     I  of  63? 

11.  I  of  48?  14.     5  of  72?  17.     |of54? 


B.     1.  How  many  are  4|  times  9  ? 


LESSON    FORTY-NINTH.  125 


l«t  Solution.  — 4§  times  9=4  times  9  +  §  of  9  =  4  times  9  +  2 
times  i  of  9.  4  times  9  =  36 ;  J  of  9  =  3,  and  2  times  3  =  6,  which, 
added  to  36  ==  42.     Hence,  4§  times  9  =  42. 

2d  Solution.— 4  times  9  =  36.  J  of  9  =  3,  and  §  of  9  must  equal 
2  times  3,  or  6,  which,  added  to  36  =  42.     Hence,  4^  times  9  =  42. 

2.  71  times  6  ?  8.  |  of  54  ==  *  times  3  ? 

3.  9 J  times  8  ?  9.  f  of  42  =  *  times  9  ? 

4.  8J  times  10  ?  10.  |  of  40  =  *  times  7  ? 

5.  6§  times  9  ?  11.  8§  times  9  =  ^  times  10? 

6.  8|  times  12  ?  12.  5|  times  12  =  *  times  8  ? 

7.  5^  times  14  ?  13,  7i  times  8  =  *  times  5  ? 

14.  I  of  30  +  f  of  56  =  ^  times  5  ? 

15.  «  of  45  +  I  of  25  =  *  times  9  ? 

16.  4  of  49  +  I  of  36  =  *  times  6  ? 

17.  3  of  72+  3  of  63  ==*  times  9? 

18.  f  of  64  +  4  of  28  =  *  times  7? 

C.  1.  If  8  pictures  cost  72  cents,  how  muchwill 
5  cost  ? 

Solution.  —  If  8  pictures  cost  72  cents,  1  picture  will  cost  J  of  72 
cents,  or  9  cents,  and  5  pictures  will  cost  5  times  9  cents,  or  45  cents. 
Hence,  if  8  pictures  cost  72  cents,  5  will  cost  45  cents. 

2.  If  7  sheep  cost  $49,  how  many  dollars  will  3  sheep  cost  ? 

3.  If  8  papers  of  candy  cost  66  cents,  how  much  will  5  papers 
cost? 

4.  If  a  girl  receives  45  merit-marks  for  9  perfect  lessons,  how 
many  will  she  receive  for  5  perfect  lessons  ? 

5.  If  3  milk  cans  will  hold  24  quarts  of  milk,  how  many  quarts 
will  7  milk  cans  hold  ? 

6.  How  much  will  3  quarts  of  molasses  cost  at  32  cents  per 
gallon  ? 

11* 


126  colburn's   first  part. 


Solution.  —  If  1  gal  or  4  qt.  cost  32  cents,  1  quart,  or  i  of  a  gal., 
wil.  cost  i  of  32  cents,  or  8  cents,  and  3  quarts  will  cost  3  times  8  cents, 
or  24  cents.  Therefore,  3  quarts  of  molasses,  at  32  cents  per  gallon, 
would  cost  24  cents. 

7.  How  mucli  will  7  qt.  of  meal  cost  at  24  cents  per  pk.  ? 

8.  How  much  will  3  gills  of  oil  cost  at  48  cents  per  qt.  ? 

9.  If  a  yard  of  cloth  is  worth  24  cents,  how  much  is  a  piece  2 
feet  in  length  worth  ? 

10.  If  1  acre  3  roods  (or  7  roods)  of  land  cost  63  dollars,  how 
many  dollars  will  1  acre  cost  ? 

11.  If  1  gal.  1  qt.  of  burning  fluid  cost  81  cents,  how  much 
will  1  qt.  cost?     How  much  will  1  gallon  cost? 

12.  If  a  pound  of  coffee  cost  42  cents,  what  will  ^  of  a  pound 
cost? 

Solution. — If  a  pound  of  coffee  cost  42  cents,  I  of  a  pound  will  cost 
I  of  42  cents,  which  is  6  cents,  and  |-  of  a  pound  will  cost  6  times  6 
cents,  or  36  cents.  Therefore,  ^  of  a  pound  of  coffee,  at  42  cents  per 
pound,  will  cost  36  cents. 

13.  If  a  man  can  perform  a  piece  of  work  in  72  minutes,  in  how 
many  minutes  could  he  perform  ^  of  it  ? 

14.  What  will  y  of  a  yard  of  linen  cost  at  64  cents  per  yard  ? 

15.  Brass  is  composed  of  6opper  and  zinc.  If  J  of  it  is  zinc, 
and  the  rest  copper,  how  many  pounds  of  copper  will  there  bo  in 
a  bar  of  brass  weighing  25  lbs.  ? 

16.  How  much  will  5|  lb.  of  sugar  cost  at  8  cents  per  lb. 

17.  How  much  will  8 J  barrels  of  flour  cost  at  6  dollars  per 
barrel  ? 

18.  How  many  square  rods  of  land  are  there  in  a  piece  9  rods 
long  and  6§  rods  wide  ? 

Solution. — Since  a  piece  of  land  1  rod  long  and  1  rod  wide  contains 
1  sq.  rd.,  a  piece  9  rods  long  and  1  rod  wide  must  contain  9  sq.  rd., 
and  a  piece  9  rods  long  and  6§  rods  wide  must  contain  65  times  9  sq. 


LESSON    rORTY- NINTH.  127 

rd.,  which  equal  60  sq.  rd.     Hence,  a  piece  of  land  9  rods  long  and  6§ 
rods  wide  conUiins  50  sq.  rd. 

19.  How  many  sq.  ft.  are  there  in  a  blackboard  12  ft.  long  and 
2 1  ft.  wide? 

20.  How  many  sq.  yd.  are  there  in  a  floor  5  yd.  long  and  i^ 
yd.  wide  ? 

21.  How  much  will  it  cost  to  paint  a  surface  8  ft.  long  and  4 J 
wide,  at  3  cents  per  sq.  ft.  ? 

22.  How  much  will  it  cost  to  plaster  a  wall  12  ft.  long  and  8J 
ft.  wide,  at  4  cents  per  sq.  ft.  ? 

23.  If  a  man  can  walk  32  miles  in  8  hours,  how  far  can  he  walk 
in  one  hour  ?     How  far  in  9J  hours  ? 

24.  If  8  tons  of  meadow-hay  cost  72  dollars,  how  much  will  5J 
tons  cost  ? 

25.  If  7  yd.  of  cloth  are  worth  49  lb.  of  butter,  how  many 
pounds  of  butter  ought  4|  yd.  of  cloth  to  be  worth  ? 

26.  Olive  has  40  picture-books,  Ella  has  |  as  many  as  Olive, 
and  Ada  has  J  as  many  as  Ella.  How  many  has  Ada?  How 
many  has  Ella  ? 

27.  John  is  J  as  old  as  his  father,  who  is  36  years  old,  and 
William  is  f  as  old  as  John.  How  old  is  John?  How  old  is 
William  ? 

28.  Edward  had  3  cents,  and  Robert  had  5.  They  put  their 
money  together  and  bought  72  filberts.  How  many  filberts  ought 
each  to  have  ? 

29.  Harriet,  Maria,  and  Caroline  sent  some  berries  to  market, 
for  which  they  received  63  cents.  Now,  if  Harriet  sent  3  quarts, 
Maria  2,  and  Caroline  4,  how  many  cents  ought  each  to  receive  ? 

30.  If  for  two  three-cent  pieces  and  2  cents  64  marbles  can  be 
bought,  how  many  marbles  can  be  bought  for  1  three-cent  piece 
and  2  cents?     How  many  for  3  three-cent  pieces  ? 


128  colburn's   first  part. 


31.  Julia's  basket  holds  3  pt.  1  gill.  If  she  can  fill  it  with  ber- 
ries in  45  minutes,  how  many  minutes  would  it  take  her  to  fill 
Susan's  basket,  which  holds  but  1  pint  ?  How  long  to  fill  Jose- 
phine's basket,  which  holds  but  1  gill  ? 


LESSON  L. 

A.  1.  Whatis  Jof  3? 

Solution.  —  i  of  3  is  |  of  1,  for  if  3  equal  things  should  be  divided 
into  4  equal  parts,  one  of  those  parts  would  equal  |  of  one  thing. 

Note. — This  may  be  illustrated  to  the  eye  by  taking  3  equal  lines 

and  dividing  them  into  4  equal  parts,  arranged  as  in 

the  figure  at  the  left.     One  part  will  then  contain  i 

—  —  ZI  HI      of  2  lines,  which,  as  will  be  seen,  is  equivalent  to  | 
of  a  line. 

2.  iof7?  4.     iof3?  6.     l^ofS? 

3.  Jof3?  5.     +ofl?  7.     J  of  4? 

B.  From  the  preceding  exercises,  it  follows  that  |,  or  §  of  1 
=  ^  of  3  ;  that  J,  or  J  of  1  =  J  of  7,  &c.  Hence,  §  of  any  num- 
ber equal  3  times  ^  of  that  number,  and  also  J  of  3  times  that 
number;  |  of  any  number  equal  4  times  ^  of  that  number,  and 
also  i  of  4  times  that  number. 

1.     What  is  ?  of  5  ? 

1st  Solution.— I  of  5  «=  7  times  J  of  5 ;  J  of  6  —  t,  and  7  times  f 
=  \s  ^  4|.     Hence  |  of  5  r=  4^. 

2d  Solution.  —  I  of  5  =  J  of  7  times  5  ;  7  times  5  =  35,  and  J  of 
35  =  4|.     Hence,  ^  of  5  =  4f. 

Note.  —  The  pupil  should  master  the  first  solution,  and  then  the 
second,  and  afterwards  be  required  to  use  in  each  example  the  one  best 
adapted  to  that  example. 


LESSON    FIFTIETH.  129 


2.  J  of  2?  4.     fofS?  6.     y\of7? 

3.  5  of  6?  6.     j^ofS?  7.     3  of  4? 

C.  1.  What  is  ^  of  6T  ? 

1st  Solution.— i  of  67  =-  J  of  64  -|-  J  of  3 ;  |  of  64  —  8 ;  J  of  3  = 
I,  which,  added  to  8  =  8f .     Hence,  i  of  67  —  8f. 

2d.  Solution.— i  of  67  =  J  of  64  -f-  i  of  3  «=  SJ. 

3d  Solution.— i  of  67  =  8f . 

Note. — The  first  and  second  solutions  are  chiefly  valuabla  as  a  pre- 
paration for  the  third. 

2.  iof43?  5.     fofl7?  8.     f^ofSQ? 

3.  ^of28?  6.     4  of  20?  9.     J  of  43? 

4.  J  of  17?  7.     I  of  80?  10.     J  of  27? 

D.  1.  i  of  52|  ? 

Solution.  —  i  of  52f  =  ^  of  48  -f  J  of  4|  ;  J  of  48  —  8 ;  J  of  4« 
or  of  3^0  -=  5^  which,  added  to  8  =  8^.    Hence,  J  of  52  2  „.  8  5. 

2.  iofl7i?  8.  J  of  17  qt.  1  pt.  ? 

3.  iof41|?  9.  lof41pk.  5qt.  ? 

4.  ;  of  66|  ?  10.  1  of  66  sq.  yd.  8  sq.  ft.  ? 
6.  |of26f?  11.  |of49bu.  2pk.? 

6.  f  of  86i  ?  12.     i  of  58  yd.  2  ft.  ? 

7.  I  of  75  ?  13.     I  of  26  wk.  4  days  ? 

E.  1.     g  of  36  =  *  times  5  ?        4.     »  of  49  ==  *  times  8? 

2.  f  of25=r  *times2?        5.     ^  of  45  =  *  times  8? 

3.  j\  of  70  ==  ^  times  5?        6.     f  of  63  =  *  times  6  ? 

F.     1.  I  of  45  =  *  times  J  of  42  ? 


130  colburn's  first  part. 


Solution.  —  |  of  45  =  8  times  l.  of  45 ;  ^  of  45  =  5,  and  8  times 
5  =  40 ;  I  of  42  =  7,  and  7  is  contained  in  40,  5 1  times.  Therefore, 
I  of  45  =  5|  times  i  of  42. 

2.  j\  of  80  =  *  times  J  of  64  ? 

3.  I  of  36  ==  *  times  J  of  24? 

4.  I  of  72  =  *  times  }  of  32  ? 

5.  I  of  40  =  *  times  |  of  12  ? 

6.  I  of  16  =  *  times  J  of  9? 

7.  ^  of  25  =  *  times  ^  of  9  ? 

8.  I  of  19  =  ^  times  J  of  7  ? 

9.  J  of  14  =  *  times  i  of  8? 

G.     1.  If  7  inkstands  cost  45  cents,  what  will  3  cost  ? 

2.  K  9  melons  cost  77  cents,  what  will  5  cost  ? 

3.  If  8  weeks*  board  cost  $27,  what  will  7  cost? 

4.  If  4  men  eat  23  pounds  of  meat  in  a  month,  how  many 
pounds  will  7  men  eat  in  the  same  time  ? 

6.  If  8  horses  eat  37  cwt.  of  hay  in  a  month,  how  much  will  5 
horses  eat  in  the  same  time  ? 

6.  If  1  pk.  of  cranberries  cost  63  cents,  how  many  cwt.  will  3 
qt.  cost? 

7.  If  a  gallon  of  burning  fluid  is  worth  79  cents,  what  are  1  qt. 
1  pt.  worth  ? 

8.  f  of  a  furlong  =  how  many  rods  ? 

Suggestion.— Since  40  rods  =  1  furlong,  |-  of  a  furlong  equal  |  of 
40  rods. 

9.  f  of  a  qr.  =  how  many  pounds  t 

10.  5  of  a  cu.  yd.  =  how  many  cubic  feet  ? 

11.  |.  of  an  hour  =  how  man^  minutes  ? 


LESSON    FIFTIETH.  131 


12.  ^  of  a  day  =  how  many  hours  ? 

13.  f  of  a  bu.  =  how  many  pk.,  qt.,  &c.  ? 


\ 


Solution. —  Since  1  bu.  =  4  pk.,  §  of  a  bu.  must  equal  f  of  4  pk., 
or  2§  pk.  But  since  1  pk.  =  8  qt,  §  of  a  pk.  must  equal  ^  of  8 
qt.,  or  5 J  qt.  Since  1  qt.  =  2  pt.,  J  of  a  qt.  must  equal  J  of  2  pt., 
or  §  of  a  pt.  Since  1  pt.  =  4  gills,  §  of  a  pt.  must  equal  §  of  4  gills, 
or  23  gills.     Therefore,  §  of  a  bu.  =  2  pk.  5  qt.  0  pt.  2§  gills. 

14.  ^  of  a  £  =  how  many  s.,  d.,  and  qr.  ? 

15.  ^  of  a  lb.  =  how  many  oz.  and  dwt.  ? 

16.  ^  of  a  ton  =  how  many  cwt.,  qr.,  lb.  ? 

17.  I  of  a  sq.  yd.  =  how  many  sq.  ft,  sq.  in.  ? 

18.  ^  of  a  lb.  =  how  many  oz.,  dwt.,  gr.  ? 

19.  I  of  a  wk.  =  how  many  da.,  h.,  &c.  ? 

20.  3  of  a  lb.  =  how  many  g.,  g.,  &c.  ? 

21.  J  of  a  £.  =  how  many  s.,  d.  qr.  ? 

22.  Frederic  and  Benjamin  gathered  some  nuts,  of  which  Fre- 
derick gathered  4  qts.,  and  Benjamin  2  qts.  They  sold  them  for 
39  cents.     How  many  cents  ought  each  to  receive  ? 

23.  Mr.  Ames  and  Mr.  Clapp  bought  the  apples  on  2  large  trees 
for  9  dollars.  Mr.  Ames  paid  5  dollars,  and  Mr.  Clapp  paid  4 
dollars.  There  proved  to  be  87  pecks  of  apples  on  the  trees. 
How  many  pecks  ought  each  to  have  ?     How  many  bushels  ? 

24.  If  4  pounds  of  rice  are  worth  24  cents,  how  many  poundo 
)f  rice  ought  to  be  given  for  7  J  pounds  of  sugar  worth  8  cents  per 
pound  ? 

25.  How  many  half-pounds  of  coffee,  worth  16  cents  per 
pound,  would  be  given  for  2J  bushels  of  oats,  worth  44  cents  per 
bushel  ? 

26.  I  gave  4  pk.  3  qt.  of  nuts,  worth  24  cents  per  peck,  for  eggs 
at  12  cents  per  dozen.     How  many  dozen  eggs  did  I  receive  ? 


132  colburn's  first   part. 


27.  I  worked  3J  weeks,  at  $18  per  week,  and  received  in  pay- 
ment $30  in  money  and  the  balance  in  shoes  at  $3  per  pair. 
How  many  pairs  of  shoes  did  I  receive  ? 

28.  James  asked  his  father  for  some  drawing-pencils,  to  which 
his  father  replied,  **  4  good  drawing-pencils  will  cost  as  much  as 
3  writing  books  at  $1  per  dozen.  Now,  if  you  will  tell  me  how 
much  6  drawing-pencils  will  cost,  I  will  buy  them  for  you.'' 
What  should  have  been  James's  answer  ? 


LESSON  LI. 

A.  1.  What  part  of  ^  is  f  ? 

Solution. — 2  \s  the  same  part  of  |  that  2  is  of  5,  which  is  |. 

What  part  — 

2.  OfAisJ?  4.     Of  2V  is  aV-  ^' 

3.  Of^is/^?         5.     Of  J  is  4?  7.     Of  f  is  J  ? 

8.  What  part  of  2|  is  10|  ? 

Solution. — 2^  =  1 ;  10 J,  and  ^  is  the  same  part  of  |  that  31  is  of  8, 
which  is  3^1  or  3J.     Hence,  lOJ  =  3^1  of  2§,  or  it  equals  3|  times  2^. 

What  part  — 

9.     Of  2  J  is  4  J  ?  14.  Of  3  da.  is  1  wk.  ? 

10.  Of  71  is  2f  ?  15.  Of  4  sq.  ft  is  1  sq.  yd. 

11.  Of9|is2|?  16.  Of  1  pk.  1  qt.  is  3  pk.  7  qt.  ? 

12.  Of  J  is  1?  17.  Of3yd.  1ft.  is8yd.  2ft.? 

13.  Of  I  is  1  ?  18.  Of  9  d.  1  qr.  is  3  d.  2  qr.  ? 

B.  1.  3  =  ^  of  what  number  ? 


2. 

9  =  1  of  *  ? 

3. 

7  =  iof*? 

4. 

3  =  Jof*? 

6. 

|  =  iof^? 

6. 

|  =  iof^? 

LESSON    FIFTY-FIRST.  133 

1st  Solution. — 3  =  ^  of  8  times  3,  or  24. 

2d  Solution. — If  3  is  ^  of  some  number,  1  of  the  number  must  be  8 
times  3,  or  24.    Hence,  8  -=  i  of  24. 


7.  f  =  itof^e? 

8.  9  J  =  i  of  *  ? 

9.  7|  =  i  of  *  ? 

10.  2bu.  3pk.  =  Jof  *? 

11.  5wk.  3da.  ==  Jof  *? 


C.     1.  17  =  I  of  what  number  ? 

Solution. — If  17  =  |  of  some  number,  I  of  that  number  must  be  i 
of  17,  which  is  5|,  and  J  of  the  number  must  be  4  times  5§,  which  is 
22t.     Therefore,  17  =  I  of  22f . 

Note.— To  prove  this,  see  if  |  of  22§  =  17. 

2.  36  =  4  of  *  ?  12.  3|  times  10  ==  -J  of  *? 

3.  32  =  |of^?  13.  91  times7  =  |of  *? 

4.  40  =  I  of  *  ?  14.  5|  times  9  =  |  of  *  ? 

5.  42=«of*?  16.  8Jtimes8  = -jPg  of *? 

6.  81  =  j\  of  *  ?  16.  8J  times  9  =  13  times  *  ? 

7.  27  =  I  of  ^  ?  17.  9J  times  6  =  2}  times  *  ? 

18.  8|  times  10  =  If  times  *  ? 

19.  3f  times  9  =  3J  times  *  ? 

10.  40  =  I  of  *  ?  20.     6}  times  8  =  1|  times  *  ? 

11.  I  of  45  =  5  of  *  ?        21.     8|  times  6  =  2J  times  *  ? 

D.  1.  If  I  of  a  gallon  of  molasses  cost  25  cents,  what  will  1 
gallon  cost  ? 

12  ~~  ~      ' -' 


134  colburn's  first   part. 


Solution. — If  £  of  a  gallon  cost  25  cents,  ^  of  a  gallon  will  cost  J 
of  25  cents,  which  is  8i  cents ;  and  J  of  a  gallon  will  cost  4  times  8J 
cents,  which  are  33^  cents.  Therefore,  1  gallon  of  molasses  will  cost 
33i  cents,  if  |  of  a  gallon  cost  25  cents. 

2.  If  I  of  a  yard  of  muslin  cost  37  cents,  "wliat  will  1  yard 
cost? 

3.  If  |-  of  a  yard  of  linen  cost  53  cents,  what  will  1  yard  cost  ? 

4.  If  J  of  a  month's  wages  amount  to  23  dollars,  what  will  1 
month's  wages  amount  to  ? 

5.  John  is  17  years  old,  and  his  age  is  |  of  his  teacher's  age. 
How  old  is  his  teacher  ? 

6.  Deborah  says  that  she  has  41  cents.  "  Then,"  said  Lavinia, 
"  you  have  only  -^^  as  many  as  I  have."  How  many  cents  had 
Lavinia  ? 

7.  David  told  George  that  ^  of  his  money  would  buy  5J  pounds 
of  raisins  at  9  cents  per  pound.  **  Then,"  replied  George-,  "  you 
have  6  cents  more  than  I  have."  How  many  cents  did  each  of 
the  boys  have  ? 

8.  Seth's  father  gave  him  a  half-dollar  to  buy  a  pound  of  tea 
with,  saying  to  him,  "4  of  a  pound  of  tea  will  cost  80  centn,  and 
if  you  will  tell  me  how  much  a  pound  will  cost,  you  may  have  the 
money  which  will  be  left  after  paying  for  the  tea."  Seth  an- 
swered correctly.  What  was  his  answer  ?  How  many  cents  would 
be  left  after  paying  for  the  tea  ? 

9.  A  farmer  gave  7f  dozen  of  eggs  at  9  cents  per  dozen  for  J 
of  a  gallon  of  oil.     What  was  a  gallon  of  the  oil  worth  ? 

10.  A  butcher  received  35  shillings  for  f  of  a  hundred  weight 
of  beef.  What  would  ho  receive  for  a  hundred  weight  at  the  same 
price.     What  would  he  receive  for  -^^  of  a  hundred  weight  ? 

11.  A  schoolmaster  being  asked  his  age,  replied,  *'  f  of  my  life 
have  been  spent  in  teaching.  I  have  taught  in  Boston  25  years, 
which  is  f  of  all  the  time  I  have  taught."     What  was  his  age  ? 


LESSON    FIFTY-FIRST.  135 


12.  If  a  yard  of  muslin  costs  70  cents,  and  ^  of  a  yard  of  mus- 
lin costs  I  as  much  as  a  yard  of  cambric,  what  will  a  yard  of 
cambric  cost? 

13.  If  a  yard  of  linen  costs  56  cents,  and  |  of  a  yard  of  linen 
costs  I  as  much  as  a  yard  of  lawn,  how  much  will  a  yard  of  lawn 
cost  ?     How  much  will  J  of  a  yard  of  lawn  cost  ? 

14.  If  Joseph  can  earn  54  cents  in  one  day,  and  it  takes  William 
y^j  of  a  day  to  earn  as  much  as  Joseph  can  earn  in  |  of  a  day, 
how  much  can  William  earn  in  one  day  ? 

15.  Mr.  Battles  gave  25  dollars  for  a  cow,  and  1|  times  what 
he  gave  for  the  cow  is  equal  to  2 J  times  what  he  gave  for  a  heifer. 
What  did  he  give  for  the  heifer  ? 

16.  After  Arthur  had  given  -|  of  his  money  for  a  blank  book, 
and  I  of  it  for  a  grammar,  he  had  18  cents  left.  How  many  cents 
had  he  at  first  ? 

17.  I  of  Mr.  Ball's  farm  is  tillage,  *  of  it  is  pasturage  and  the 
rest,  3  acres,  is  orchard.     How  many  acres  does  he  own  ? 

18.  William  and  Henry  were  trying  to  puzzle  each  other  about 
their  ages.  William  told  Henry  that  he  had  spent  J  of  his  life  in 
Philadelphia,  f  of  it  in  New  York,  and  the  rest  in  Boston,  and 
that  he  had  lived  5  years  more  in  New  York  than  in  Boston. 
Henry  found  out  his  age.     What  was  it  ? 

19.  Henry  then  said  to  William,  *'I  have  lived  in  Hartford, 
Providence,  and  Boston.  I  spent  -^^  of  my  life  in  Hartford ;  I 
lived  in  Providence  twice  as  long  as  in  Hartford,  and  have  livei 
in  Boston  2  years  more  than  3  times  as  long  as  in  Providence." 
William  found  out  Henry's  age.     What  was  it  ? 


136  colburn's  first  part. 


LESSON  LII. 

A.  1.  What  is  the  effect  of  multiplying  the  nu- 
merator of  the  fraction  j%  by  3  ? 

Ans. — Multiplying  the  numerator  of  the  fraction  y*^  by  3,  gives 
1^  for  a  result,  which  expresses  3  times  as  many  parts,  each  of 
the  same  .size  as  before,  and  is,  therefore,  3  times  as  large. 
Hence,  multiplying  the  numerator  of  j\  by  3,  multiplies  the  frac- 
tion by  3. 

What  is  the  effect  of  multiplying  the  numerator 
of— 

2.  T\by2?  4.     4  by  3?  6.     4  by  5? 

3.  2\by6?  6.     8  by  7?  7.     |  by  9  ? 

8.  What  is  the  effect  of  dividing  the  numerator 
of  the  fraction  j|  by  6  ? 

Ans. — Dividing  the  numerator  of  ||  by  6,  gives  j^j  for  a  result, 
which  expresses  J  as  many  parts,  each  of  the  same  size  as  before, 
and  is  therefore  J  as  large.  Hence,  dividing  the  numerator  of 
If  ^y  6,  divides  the  fraction  by  6. 

Note. — Tho  first  of  the  above  solutions  is  equivalent  to  "3  times 
y^y  =  XZj  just  as  3  times  4  apples  =  12  apples;"  and  the  second  is 
equivalent  to  "  J  of  j  |  ==  J^^,  just  as  J  of  12  apples  =  2  apples." 
They  are  necessary  as  a  preparation  for  the  exercises  which  follow. 

What  is  the  effect  of  dividing  the  numerator  of — 

9.     14  by  5?  11.     I  by  4?  13.     |fby7? 

10.     I?  by  2?  12.     §by8?  14.     ||by9? 


LESSON    FIFTY-SECOND.  137 


Hence,  multiplying  the  numerator  of  a  fraction  multiplies  the  frac- 
tion, and  dividing  the  numerator  divides  the  fraction, 

B.  To  THE  Teacher. — Should  the  pupils  find  any  difficulty  in  under- 
standing the  following  exercises,  illustrations  should  be  given  by  divi- 
ding visible  objects,  such  as  apples,  lines,  <fcc.,  into  various  kinds  of 
fractional  parts. 

From  the  nature  of  fractional  partr,  it  follows  that  — 

1st.  The  larger  the  number  of  fractional  parts  into  which  any 

unit  is  divided,  or  which  it  takes  to  equal  that  unit,  the  smaller 

each  part  will  be. 

2d.  The  smaller  the  number  of  fractional  parts  into  which  any 

unit  is  divided,  or  which  it  takes  to  equal  that  unit,  the  larger 

each  part  will  be. 

1.  Whicli  parts  are    larger    in    size,  halves   or 

fourths  ? 

Ans.  — Halves,  because  it  takes  a  less  number  of  them  to  equal 
a  unit. 

Which  parts  are  larger  in  size  — 

2.     Halves  or  thirds  ?  6.     Halves  or  tenths  ? 

8.     Fourths  or  eighths  ?  6.     Fourths  or  twelfths  ? 

4.     Thirds  or  ninths  ?  7.     Fifths  or  twentieths  ? 

C.     1.  Each  half  equals  how  many  sixths  ? 

1st  Form  of  Answer.  —  Each  half  equals  J  of  6  sixths,  which 
is  3  sixths.     Hence,  each  half  equals  3  sixths. 

2d  Form  of  Answer.  —  Each  half  equals  3  sixths,  for  if  a  unit 
should  be  divided  into  6  equal  parts,  J  of  the  unit  would  contain 
3  of  them. 


138  colburn's  first  part. 


2.  Each  half  equals  how  many  fourths  ? 

3.  Each  third  equals  how  many  ninths  ? 

4.  Each  fourth  equals  how  many  twelfths  ? 

5.  Each  fifth  equals  how  many  tenths  ? 

6.  Each  sixth  equals  how  many  eighteenths  ? 

D.     From  the  foregoing  exercises  we  may  infer  that  — 

1st.  Multiplying  the  number  of  fractional  parts  into  which  a  unit 

is  divided,  or  which  it  takes  to  equal  a  unity  divides  each  part. 

2d.  Dividing  the  number  of  parts  into  which  a  unit  is  divided,  or 

which  it  takes  to  equal  a  unit,  multiplies  each  part. 

1.  What  is  the  eifect  of  multiplying  the  denomi- 
nator of  the  fraction  |  by  4  ? 

Ans.  —  Multiplying  the  denominator  of  the  fraction  J  by  4, 
gives  -^^  for  a  result,  which  expresses  the  same  number  of  parts 
each  J  as  large  as  before.  Hence,  -^^  =  J  of  J,  or  multiplying 
the  denominator  of  ^  by  4,  has  divided  the  fraction  by  4. 

What  is  the  effect  of  multiplying  the  denominator 
of— 

2.  §by5?  4.     fby2?  6.     |by3? 

3.  §by6?  5.     4  by  3?  7.     fby4? 

What  is  the  effect  of  dividing  the  denominator  of 

Ans. — Dividing  the  denominator  of  the  fraction  -f^  by  2,  gives  | 
for  a  result,  which  expresses  the  same  number  of  parts  each  twice 
as  large  as  before.  Hence,  |  =  two  times  ^^,  or  the  fraction  ^- 
has  been  divided  by  2. 


LESSON    FIFTY-SECOND.  139 

What  is  the  effect  of  dividing  the  denominator  — 
9.     Of  5  by  3?        11.     Of  2^  by  8?  13.     Of  ^^  by  7? 

10.     Of  I  by  2?        12.     Of  2^^  by  5?  14.     Of  |J  by  9  ? 

E.  The  numerator  and  denominator  are  called  terms  of  the 
fraction. 

I.  What  is  the  effect  of  multiplying  both  terms 
of  I  by  6  ? 

Ans. — Multiplying  both  terms  of  the  fraction  f  by  6,  gives  -}-| 
for  a  result,  which  expresses  6  times  as  many  parts,  each  J  as 
large  as  before.     Hence,  the  value  of  the  fraction  is  not  altered, 

What  is  the  effect  of  multiplying  both  terms  of — 

2.  J  by  3?  6.     /ffby3?  8.     |by8? 

3.  J  by  2?  6.     I  by  4?  9.     /^byS? 

4.  I  by  4?  7.     fby7?  10.     fby6? 

II.  What  is  the  effect  of  dividing  both  terms  of 
if  by  5. 

Ans. — Dividing  both  terms  of  the  fraction  -15  by  5,  gives  |  for 
a  result,  which  expresses  J  as  many  parts  each  5  times  as  large 
as  before.     Hence,  the  value  of  the  fraction  is  not  altered,  or 

15  3 

25  —  5- 

What  is  the  effect  of  dividing  both  terms  of — 

12.  -i%by3?  15.     T\by3?  18.     i4by7? 

13.  4  by  4?  16.     /^by3?  19.     J|  by  5  ? 

14.  -{-«by5?  17.     i|by7?  20.     f  J  by  9  ? 


140  colburn's   first  part. 


F.     From  the  foregoing,  it  appears  that  — 

1.  Multiplying  the  numerator  multiplies  the  fractiony  hy  multiply- 
ing the  number  of  parts  considered^  without  affecting  their  size. 

2.  Dividing  the  numerator  divides  the  fractiony  hy  dividing  the  num- 
ber of  parts  considered^  without  affecting  their  size. 

3.  Multiplying  the  denominator  divides  the  fractiony  by  dividing 
each  party  without  affecting  the  number  of  parts  considered. 

4.  Dividing  the  denominator  multiplies  the  fraction,  by  multiplying 
each  party  without  affecting  the  number  of  parts  considered. 

6.  A  fraction  may  be  multiplied  either  by  multiplying  the  numerator 
or  by  dividing  the  denominator. 

6.  A  fraction  may  be  divided  either  by  dividing  the  numerator  or 
by  multiplying  the  denominator. 

7.  Multiplying  both  numerator  and  denominator  of  a  fraction  by 
the  same  number  both  multiplies  and  divides  the  fraction  by  that  num- 
ber, andy  therefore,  does  not  alter  its  value. 

8.  Dividing  both  numerator  and  denominator  of  a  fraction  by  the 
same  number,  both  divides  and  multiplies  the  fraction  by  that  number, 
and,  thereforey  does  not  alter  its  value. 


LESSON  LIII. 

A.  A  fraction  is  said  to  be  in  its  Lowest  Terms  when  its 
numerator  and  denominator  are  the  smallest  entire  numbers  which 
will  express  its  value. 

When  a  fraction  is  in  its  Lowest  Terms,  there  is  no  entire  num- 
ber greater  than  one  which  will  divide  both  numerator  and  deno- 
minator without  a  remainder. 


LESSON    FIFTY-THIRD.  141 


Hence,  a  fraction  may  he  reduced  to  its  lowest  terms  hy  dividing 
both  numerator  and  denominator  by  the  same  numbers, 

1.     Reduce  sf  to  its  lowest  terms. 

Solution.  —  Both  terms  of  jL|  can  be  divided  by  6.  Dividing 
them,  gives  |  for  a  result,  which  expresses '  i  as  many  parts,  each 
6  times  as  large  as  before.  Hence,  ±|  —  |,  and  as  |  admits  of  no  fur- 
ther reduction,  it  is  the  fraction  required. 

The  number  by  which  we  divide  in  reducing  fractions  to  their 
lowest  terms,  are  said  to  be  canceled.  Thus,  in  the  first  solu- 
tion, we  canceled  the  factor  6 ;  in  the  second,  we  canceled  the 
factor  2,  and  then  the  factor  3. 

Note.  —  The  pupil  should  not  only  master  the  explanation,  but 
should  also  learn  to  give  the  results  without  the  explanation.  Let  him 
also  observe  that  a  fraction  can  always  be  reduced  to  its  lowest  terms 
by  dividing  both  terms  by  their  greatest  common  divisor. 

Reduce  each  of  the  following  fractions  to  its  lowest 
terms : — 

2.  tV  5.    if.  8.    Jf.  11.    A8. 

3.  j%.  6.    if.  9.    If.  12.    4|. 

4.  ^\.  7.    J|.  10.    if.  13.    ^, 

B.  A  fraction  is  sometimes  expressed  by  the  factors  of  its 
numerator  and  denominator. 

4X9 
Example.— The  fraction — 1_     which  may  be  read,  "  The  fraction 
15  X  8, 

having  4  times  9  for  its  numerator,  and  15  times  8  for  its  denomina- 
tor,'^ or,  "  The  fraction  4  times  9,  divided  by  15  times  8." 

Such  fractions  should  be  reduced  to  their  lowest  terms  before 
muhiplying  their  factors  together. 

Reduce  ^^      ^  to  its  lowest  terms. 

Solution.— Canceling  4  from  the  factor  4  of  the  numerator  and  S 


142  colburn's  first  part. 


of  the  denominator,  gives  1  in  place  of  the  former,  and  2  in  place  of 

the  latter.*     Cancelling  3  from  the  factor  9  in  the  numerator,  and  15 

of  the  denominator,  gives  3  in  place  of  the  former,  and  5  in  place  of 

the  latter. ■}•     As  no  further  division  can  be  made,  we  multiply  the  re- 

4X9        1X3        3 
maining  factors  together,  which  gives == =  — 

In  writing  out  the  work,  it  is  customary  to  draw  a  line  through 
the  numbers  from  which  factors  have  been  canceled,  and  to  write 
the  quotients  above  the  dividends  of  the  numerator,  and  below 
those  of  the  denominator. 

Reduce  each  of  the  following  to  its  lowest  terms  : 

^6X4  Q      28  X  36  ^^    3x4x6 

'     24  X  14  *    6  X  8  X  3 

^     49  X  25  ^j     4  X  0  X  21 

*     35  X  35  *    14  X  G  X  3 

«      2x3x4        ..,     15  X  7X  13 


9X8 

10 

X 

9 

21 

X 

6 

12 

X 

7 

35 

X 

18 

16 

X 

18 

12. 


6x5x8  13x35x9 

9      4x7x9        ^3     24  X  18  X  25 

45  X  24  '7x6x8  '      36  x  45  x  56. 


LESSON  LIV. 
A.     1.  How  can  J  of  a  fraction  be  found  ? 

Ans. — J  of  a  fraction  can  be  found  by  dividing  its  numerator 
by  2  ;  or  by  multiplying  its  denominator  by  2.  For,  dividing  the 
numerator  by  2,  will  give  for  a  result  a  fraction  expressing  J  as 
many  parts,  each  of  the  same  size  as  those  of  the  given  fraction  ; 


*This  makes  the  fraction  express  i  as  many  parts,  each  4  times  as 
large  as  before,  and  hence  does  not  alter  its  value. 

f  This  makes  the  fraction  express  J  as  many  parts,  each  3  times  as 
large  as  before,  and  therefore  does  not  alter  its  value. 


LESSON    PIFTY-FOURTH.  143 


or,  dividing  the  denominator,  will  give  for  a  result  a  fraction 
expressing  the  same  number  of  parts,  each  J  as  large  as  those  of 
the  given  fraction. 

Note. — The  statement  of  the  reasons  should  not  be  omitted  till  it  i« 
certain  that  the  pupil  fully  understands  them. 

State  the  method  of  finding  — 

2.  J  of  a  fraction.                           5.  -jJ^  of  a  fraction. 

3.  )j  of  a  fraction.                            6.  J  of  a  fraction. 

4.  I-  of  a  fraction.                            7.  J^  of  a  fraction. 

B.  1.  What  is  the  effect  of  multiplying  the  nu- 
merator of  a  fraction  by  3,  and  the  denominator  by 
4? 

Ans. — Multiplying  the  numerator  of  a  fraction  by  3,  and  the 
denominator  by  4,  gives  for  a  result  f  of  the  original  fraction,  for 
it  gives  3  times  as  many  parts,  each  J  as  large  as  before. 

What  is  the  effect  of  multiplying  the  numerator 
of  a  fraction  — 

2.  By  4,  and  the  denominator  by  7  ? 

3.  By  8,  and  the  denominator  by  3  ? 

4.  By  11,  and  the  denominator  by  6  ? 

5.  By  12,  and  the  denominator  by  10  ? 

6.  By  24,  and  the  denominator  by  17  ? 

C.  1.  How  can  |  of  a  fraction  be  found  ? 

1st  Method. — ^  of  a  fraction  can  be  found  by  getting  |  of  the  nu- 
merator for  a  new  numerator,  without  altering  the  denominator. 

2d  Method.  —  f  <>^  *  fraction  can  be  found  by  multiplying  the  nu- 
merator by  5  and  the  denominator  by  6. 


144 


COLBURN    S    FIRST    PART. 


Note.  —  The  pupil  should  observe  that  the  first  method  gives  |  as 
many  parts  of  the  same  size  as  before,  and  that  the  second  gives  5 
times  as  many  parts,  each  ^  as  large  as  before.  He  should  observe, 
further,  that  the  first  method  is  identical  with  that  of  Lesson  L. 

Explain  the  methods  of  finding  — 


2. 

1^  of  a  fraction. 

6. 

1  of  a  fraction. 

3. 

^  of  a  fraction. 

7. 

IJ  of  a  fraction. 

4. 

y^  of  a  fraction. 

8. 

j^  of  a  fraction. 

6. 

y'*j  of  a  fraction. 

9. 

J  of  a  fraction 

D. 

1.  What 

is  \ 

Ofy^? 

2. 

J  off? 

1 

3 

2 
6.       }of}? 

10.      |of||» 

3. 

tofA^ 

7.     fjof§|? 

11.    /,off? 

4. 

T^.0f|? 

8.     Ifofil? 

12.      |of|f? 

5. 

|ofJ? 

9.       ioff? 

18.    ^\otJ^1 

E.     1.  What  is  the  product  of  f  times  ^f  ? 

3 

Solution.-  f  times  if  - 1  of  if  ^  ^  ^^  ^  tV 

2         5 


9  X  28      6 


2.  What  is  the  product  of  y^  x  ||  ? 

1st  Solution.— y\  X  f f ,  or  fV  times  f|=A  of  §|=     jj^^^ 

2d.  Solution.  —  /^  X  ff,  or  j\  multiplied  by  f|,  -=  ||of  ^9^=» 

28  X    9_  6 

33  X  14      ^^' 


LESSON    FIFTY-FOURTH.  145 

Note. — The  slight  difference  between  the  first  and  second  solution 
results  from  the  different  reading  of  the  sign  of  multiplication.  We 
recommend  the  first  as  being  the  most  simple. 


5.       |X   ^L?         8.     ^fXi??        11.        4  X  JX   ax  VV? 

12.     4J  X  21? 

Solution.  — 4ix  2|  =  |  X  y==  V  =  ^2^* 

13.  2|x4J?  15.     2JX3J?  17.     If  X  IJ? 

14.  5JxH?  16.     4fxl-|?  18.     8JX7J? 


G.     1.  f  =  I  of  what  number  ? 

Solution. — If  f  =  «  of  some  number,  i  of  that  number  must  be  i 

3 

of  i,  which  is    =  J ;  and  ^  of  the  number  must  be  7  times  J, 

which  are  ^.     Hence,  |  =  5  of  f . 


2.  |=:40f^-?  6.  ^==:^of^t 

3.  |4  =  j%  of  ^-  ?  ^.     3J  =  2}  times  ^  ? 

4.  /g  =  1^  times  -)f  ?  S.     2J  =  2f  times  *  ? 
6.      i=2i  times  *  ?                9.     4  J  =  f  of  *  ? 

H.     1.  How  much  will  f  of  a  yard  of  cloth  cost  at  |  of  a  dollar 
per  yard  ? 

2.  How  much  will  f  of  a  quart  of  filberts  cost  at  |  of  a  dime 
per  quart  ? 

3.  George  gathered  |  of  a  bushel  of  cranberries,  and  sold  f  of 
what  he  gathered.     What  part  of  a  bushel  did  he  sell  ? 

4.  Rufus  earned  J  of  a  dollar,  and  then  spent  |  of  what  he  had 
earned.     What  part  of  a  dollar  did  he  spend  ? 

j3  - 


146  colburn's   first   part. 


5.  If  1  pound  of  tea  is  worth  |  of  a  dollar,  what  is  ^  of  a  pound 
w  or  til  ? 

6.  If  a  man  can  hoe  |  of  an  acre  of  corn  in  1  day,  what  pnrt 
of  an  acre  can  he  hoe  in  J  of  a  day  ? 

7.  If  5  pounds  of  soap  cost  J  of  a  dollar,  what  will  one  pound 
cost? 

8.  If  4  oranges  cost  J  of  a  dollar,  what  will  1  orange  tost ' 
What  will  3  cost  ? 

9.  If  6  pine  apples  cost  J  of  a  dollar,  what  will  6  cost  ? 

10.  If  5  lb.  of  coffee  cost  |  of  a  dollar,  what  will  10  lb.  cost? 

11.  If  §  of  a  yard  of  silk  velvet  cost  5 J  dollars,  what  will  1 
yard  cost  ? 

Solution. — If  §  of  a  yard  of  silk  velvet  cost  5J  dollars,  ^  of  a  yard 
will  cost  i  of  5i  dollars.  ^  of  5i  =  i  of  4  -[-  i  of  U,  or  of  |,  which  is 
2|.  If  J  of  a  yard  cost  2§  dollars,  |  of  a  yard  will  cost  3  times  2^ 
dollars,  which  are  7J  dollars. 

12.  If  I  of  an  acre  cost  30J  dollars,  what  will  1  acre  cost  ? 

13  If  J  of  a  cask  of  oil  is  worth  64 J  dollars,  what  is  the  cask 
worth  ? 

14.  If  2 J  cords  of  wood  are  worth  $18J,  what  is  1  cord  worth? 

15.  If  a  wood-cutter  can  cut  6 J  acres  of  wood  in  2J  days,  how 
much  can  he  cut  in  1  day? 

16.  If  Rufus  can  shell  2f  bushels  of  corn  in  1  hour,  how  many 
bushels  can  he  shell  in  2 J  hours  ? 

17.  If  Rufus  can  shell  7J  bushels  of  corn  in  2J  hours,  how  many 
bushels  can  he  shell  in  1  hour  ? 

18.  If  Albert  can  walk  13 J  miles  in  4  hours,  how  far  can  he 
walk  in  1  hour  ?     How  far  in  2  J  hours  ? 

19  If  Albert  can  walk  7 J  miles  in  2 J  hours,  how  far  can  he 
walk  in  1  hour  ?     How  far  in  4  hours  ? 


LESSON    FIFTY-FOURTH.  147 


20.  When  4^  bushels  of  corn  can  be  bought  for  $2|,  how  many 
bushels  can  be  bought  for  1  dollar?  How  many  for  y^^  of  a 
dollar  ? 

21.  When  1|J-  bushels  of  corn  can  be  bought  for  y^^  of  a  dollar, 
how  many  bushels  can  be  bought  for  1  dollar  ?  How  many  for 
$2|? 

22.  If  J  of  a  yard  of  linen  is  given  in  exchange  for  f  of  a  yard 
of  silk  worth  |  of  a  dollar  per  yard,  what  ought  the  linen  to  be 
worth  per  yard  ? 

23.  If  I  of  a  yard  of  silk  is  given  in  exchange  for  |  of  a  yard 
of  Unen  worth  f  of  a  dollar  per  yard,  what  ought  the  silk  to  be 
worth  per  yard  ? 

24.  What  part  of  1  rod  is  4  yd.  2  ft.  1|  in.  ? 

Solution. — Since  1  in.  =j^^  of  a  foot,  1|  in.  or  1^2  of  an  inch  must 
equal  ^  of  ^^  of  a  ft.  =  1  ft.,  to  which,  adding  the  2  ft.,  gives  21 
ft,,  or  y  ft.  Since  1  ft.  =  J  of  a  yd.,  y  of  a  ft.  must  equal  y  of  i  of 
a  yd.=  I  yd.,  to  which,  adding  the  4  yd.,  gives  4-|  yd.  =  3_3  yd.  Since 
1  yd.  =  2^  of  a  rd.  ^^  of  a  yd.  must  equal  "^^^  of  Jt  rd.  =  |  rd. 
Hence,  4  yd.  2  ft.  1|  in.  =  ^  of  a  rod. 

Prove  by  Solution  to  13th  problem,  page  131. 

25.  What  part  of  1  bu.  is  3  pk.  1  qt.  1}  pt.  ? 

26.  Of  1  gal.  is  2  qt.  1  pt.  26  gi.  ? 

27.  Of  1  lb.  is  8  oz.  14f  dr.  ? 

28.  Of  1  wk.  is  5  da.  10  h.  40  m.  ? 

29.  Of  1  £.  is  2  s.  10  d.  1^  qr. 

30.  Of  lib.  is  9  §.45.29.  8gr.? 

31.  Of  1  lb.  is  4  oz.  5  dwt.  17J-  gr.? 

32.  Of  1  T.  is  2  cwt.  0  qr.  22  lb.  3  oz.  8f  02. 


148           colburn's  first   part. 

LESSON  LV. 

A.     1.1  =  how  many  times  I  ? 

Solution. — 1  =  |,  and  |  =  5  times  ^.     Hence,  1  «=  5  times  i. 

2.  What  is  the  quotient  of  1  -  i  ? 

Solution.— 1  =  .7,  and  ;f  -  *-  i  =  7  -f-  1  =  7.    Hence,  1—1  =  7. 

3.     l  =  ^timesj?                                  7.     1 -r  J? 

4.     l  =  *timesi?                                 8.     l-f-JL? 

5.     1  =  *  times  -^L  ?                              9.     1  -f-  4  ? 

6.     1  =  4t  times  J^?                              10.     1 -^  i? 

Inference.— Since  1  =  2  times  i,  =  3  times  J,  &c.,  it  follows  that 
there  will  be  2  times  as  many  halves,  3  times  as  many  thirds,  <fcc.,  as 
there  are  times  1  in  any  number. 

B.     1.  5  =  *  times  J? 

Solution. — Since  5  contains  1,  5  times,  it  must  contain  J,  3  times  5 
times,  or  15  times.     Hence,  5  =  15  times  J. 

2.  What  is  the  quotient  of  8  -^  J  ? 

Solution. — Since  the  quotient  of  8  -r-  1  =  8,  the  quotient  of  8  -r  J 
must  equal  3  times  8  or  24.     Hence,  8  -7-  J  =  24. 

3.     7  =  ^timesi?                        9.      8-f--L? 

4.     3  =  ^  times  ^  ?                       10.       2  -f-  J-  ? 

5.     5  =  *tiiPes^V-                      11-     10 -r  J? 

6.     9  =  *  times  J?                       12.      4~J? 

7.     4  =  *  times  J  ?                       13.     12  -^  ^  ? 

8.     6==^  times  J?                        14.       9-rJ? 

LESSON    FIFTY-FIFTH.  149 


C.  From  all  the  preceding  exercises,  it  must  be  obvious  that 
the  quotient  of  a  number  divided  by  1  equals  that  number.  Thus 
3  A.  1  =  3  ;  7  -r  1  =  7,  &c.     So  f  —1  =  J;  | -^-  1  =  |,  &c. 

1.  I  =  how  many  times  J  ? 

Solution.  —  Since  |  contains  1,  |  times,  it  must  contain  i,  4  times 
?  times,  which  are  ^^  times  ==  2|  times.     Hence,  3  ___  23  times  I, 

2.  What  IS  the  quotient  of  f  -^  7  ? 

Solution. — Since  the  quotient  of  §  divided  by  1  =  f ,  the  quotient 
of  I  divided  by  ^  must  equal  7  times  f,  which  are  y  =  4^.  Hence, 
A  -^  1  =  u_ 


3.       4  =  ^  times  J  ?  8.     |  -f-  J 


4.  j\  =  *  times  -J  ?  9. 

5.  §  =  ^  times  i  ?  10. 

6.  f  ==  -sf  times  J?  11. 

7.  I  =  -x-  times  J?  ,  12. 


^? 


D.  From  the  nature  of  division,  it  is  obvious  that,  while  the 
dividend  remains  the  same,  the  larger  the  divisor  is,  the  smaller  will 
be  the  quotient,  and  the  smaller  the  divisor  is,  the  larger  will  be 
the  quotient. 

Thus  the  quotient  of  8  divided  by  2  is  4,  which  is  J  of  the  quo- 
tient of  8  divided  by  1 ;  the  quotient  of  15  -f-  3  is  5,  which  is  J 
of  the  quotient  of  15  divided  by  1.  So  the  quotient  of  a  number 
divided  by  |  must  be  J  of  its  quotient  divided  by  j  ;  the  quotient 
of  a  number  divided  by  -?  must  be  J  of  its  quotient  divided  by  J, 
&c.,  &c. 

1.    8  =  *  j? 

Solution.— Since  8  contains  -J,  5  times  8  times,  it  must  contain  3 
i  of  5  times  8  times,  or  |  of  8  times,  which  are  13J  times.  Hence, 
8  =  13J  times  5. 

13*  ■  ■ 


150  colburn's   first  part. 


2.  What  is  the  quotient  of  4  -r-  |  ? 

Solution.— Since  the  quotient  of  4  divided  by  l  =  7  times  4,  the 
quotient  of  4  divided  by  3  must  equal  i  of  7  times  4,  or  1  of  4,  which 
is  9J-    Hence,  4  H-  ^  =  9^. 

3.  7  =  ^^  times  f  ?  9.  8  -f-  /^  ? 

4.  5  =:  If  times  |  ?  10.  2  -^  JL  ? 

5.  8  =  *  times  J  ?  11.  6  -f-  §  ? 

6.  1  =  *  times  |  ?  12.  1  -i-  |  ? 

7.  1  =  *  times  J  ?  13.  1  _i.  |  ? 

8.  4  =  *  times  ^t  14.  9  ~  4  ? 

E.     1.  §  =  how  many  times  |  ? 

Solution.— Since  |  contains  ^,  7  times  |  times,  it  must  contain  3 
J  of  7  times  |  times  or  |  of  |  times  =  i^  times.  Hence,  2^14 
times  3. 

2.  What  is  the  quotient  of  f  -^  4  ? 

Solution. — Since  the  quotient  of  |  divided  by  l  =  7  times  |,  the 
quotient  of  |  divided  by  «  must  equal  J  of  7  times  |,  or  1  of  |,  which, 
by  canceling  the  factor  3,  equals  ^.    Hence,  |  -^  ^  __  j^ 

3.  |  =  *times/g?  8.       |-f-f? 

4.  }  =  ^times|?  9.     91  ~- |? 
6.     f  =  *  times  /^ ?                   10.     /^  -r  |? 

6.  5  =  *  times  I? 

7.  f  =  *  times  -^p  ? 

F.  The  preceding  exercises  and  solutions  make  it  evident  that 
the  quotient  of  a  number  divided  by  f  =  J  of  3  times  the  num- 
ber sr=  1  of  the  number ;  the  quotient  of  a  number  divided  by 
i  =  J  of  9  times  the  number  =  |  of  the  number,  and  generally 
that  — 


LESSON    FIFTY-FIFTH.  161 


The  quotient  of  a  number  divided  by  a  fraction  equals  the  product 
of  that  number  multiplied  by  the  fraction  inverted. 

1.  What  is  the  quotient  of  ^  ~-  2|  ? 

Ans.-i-  -^  2f  =  4  ^  I  =  4  >^  t  =  i. 

2.  4-7-lJ?  5.     2i~4J?  8.     8J~4J? 

3.  |~7i?  6.     f|~3i?  9.     /,~|? 

4.  5i~-4f?  7.     93-7-14?  10.     8i-r5|? 

G.     Examples  in  division  of  fractions  sometimes  appear  in  the 
form  of  fractions.     They  are  then  called  complex  fractions. 

Illustration.  —  _  which  equals  4§  -^  3  \. 

A  complex  fraction,  then,  has  a  fraction  in  one  or  both  its 
terms.     They  may  be  explained  after  the  following  model :  — 

4| 

—  expresses  the  value  of  4|  equal  parts  of  such  kind  that  3  J 

of  them  will  equal  a  unit. 

Complex  fractions  may  be  reduced  to  simple  ones  by  merely 
performing  the  indicated  division. 

Thus :  -I  =  4f  -■  31  =  1 4  -^  1 6  _  ^^  X    5        35       ^ 

Reduce  each  of  the  following  to  simple  fractions : 
1.     II.  3.     !i  5      ^^ 

2  J 


3. 

Si 
41 

4. 

^ 

2.     !l.  4.     £i.  6. 

2|  8J  li 

H.     1.  How  many  melons  at  f  of  a  dime  each  can  be  bought 
for  5  dimes  ? 


152  colburn's  first  part. 

Solution.— Since  1  melon  can  be  bought  for  |  of  a  dime,  as  many 
melons  can  be  bought  for  5  dimes  as  there  are  times  |  in  5,  which  are 
I  of  5  times  or  2_o  times  ==  6|  times.  Hence,  G§  melons  at  |  of  a  dime 
a-piece  can  be  bought  for  5  dimes. 

2.  How  many  bushels  of  corn  at  f  of  a  dollar  per  bushel  can 
be  bought  for  7  dollars  ? 

3.  How  many  hours  will  it  take  a  scholar  who  learns  ^  of  a 
page  per  hour  to  learn  3  pages  ? 

4.  When  tea  is  worth  J  of  a  dollar  per  lb.,  how  many  pounds 
can  be  bought  for  $5. 

5.  If  a  man  can  walk  -^^  of  a  furlong  in  1  minute,  how  many 
minutes  will  it  take  him  to  walk  -J  of  a  furlong? 

G.  If  a  man  can  gather  |  of  the  apples  on  a  certain  tree  in  1 
hour,  how  many  hours  will  it  take  him  to  gather  -f-^  of  them  ? 

7.  Edward  divided  |  of  a  rood  of  land  into  flower-beds,  each 
containing  -^^  of  a  rood.     How  many  beds  did  he  make  ? 

8.  A  man  who  had  $5,  gave  J  of  his  money  for  grass  seed  at 
$2i  per  bushel.     How  many  bushels  did  he  buy  ? 

9.  How  many  pounds  of  pearlash  at  -^-^  of  a  dime  per  pound 
can  be  bought  for  §  of  a  pound  of  chocolate  at  3f  dimes  per  lb.? 

10.  If  Josephine  can  learn  J  of  a  lesson  in  an  hour,  how  many 
hours  will  it  take  her  to  learn  1  lesson  ? 

11.  Albert  has  a  cord  28  feet  long,  which  he  wishes  to  cut  into 
pieces  each  2|  feet  long.     How  many  pieces  will  it  make  ? 

12.  A  man  who  liad  but  $9,  invested  f  of  his  money  in  cloth 
at  IJ  dollars  per  yard,  and  the  rest  of  it  in  cloth  at  1 J  dollars  per 
yard.     How  many  yards  of  each  kind  did  he  buy  ? 

13.  When  a  bushel  of  potatoes  can  be  bought  for  |  of  a  dollar, 
how  many  bushels  of  potatoes  can  be  bought  for  9  bushels  of 
corn  at  i  of  a  dollar  per  bushel  ? 


LESSON    FIFTY-SIXTH.  153 


14.  How  many  bottles,  eacli  holding  ^  of  a.  quart,  can  be  filled 
from  f  of  a  gallon  of  wine  ? 

15.  A  farmer  has  2^  tons  of  hay  in  one  stack,  and  3|  tons  in 
another.  He  carries  it  to  market  in  loads  each  weighing  1|  tons. 
How  many  loads  will  both  stacks  make  ? 

16.  How  many  tiles  f  of  a  foot  long  and  i  of  a  foot  wide  will  it 
take  to  cover  18  sq.  ft.  of  surface  ? 

17.  How  many  square  yards  in  a  floor  12  feet  long  and  9  feet 
wide,  and  how  many  yards  of  carpeting  f  of  a  yard  wide,  will  it 
take  to  cover  it  ? 

18.  When  $1  is  received  for  f  of  a  sq.  ft.  of  land,  how  many 
dollars  will  be  received  for  a  strip  16  feet  long,  and  j^^  o^  ^  ^^ot 
wide  ? 


LESSON  LVI. 

A.     1.  f  =  how  many  twelfths  ? 

Solution. — Since  1  =:  l|,  S  of  1  must  equal  |  of  {-|,  which  are  ^^. 
Hence,  S  =  ^\. 

In  a  similar  manner  reduce  — 

9.      f  and  J  to  twelfths. 

10.  I  and  |-  to  thirty-sixths. 

11.  f  and  -p^  to  fortieths. 

12.  -^^  and  f  to  twenty-fourths. 

13.  -^^  and  :^^  to  sixtieths. 

14.  I  and  l  to  fifty -sixths. 

15.  |,  ^  and  rp^  to  seventieths. 

Let  the  pupil  now  solve  the  above  questions  by  the  following 
form : — 


2. 

f  to  sixths. 

3. 

J  to  eighths. 

4. 

1  to  tenths. 

5. 

J  to  twentieths. 

6. 

5  to  forty-fifths. 

7. 

1  to  twenty-firsts. 

8. 

IJ  to  thirty-sixths. 

154  colburn's   first   part. 

Solution  to  problem  1st. — Since  the  required  denominator,  12,  is  3 
times  the  given  denominator,  4,  we  multiply  both  terms  of  f  by  3, 
which  gives  |  ==  -j9-. 

B.  1.  Fractions  have  a  COMMON  DENOMINATOR  when  they  have 
the  same  denominator. 

Illustration. — |  and  J  do  not  have  a  common  denominator,  but 
i,  If  and  1  have  the  common  denominator  9. 

2.  Fractions  having  different  denominators  can  be  reduced  to 
a  common  denominator,  t.  e.,  to  equivalent  fractions  having  a 
common  denominator.  This  is  illustrated  in  the  last  7  examples 
under  A.  • 

3.  In  reducing  fractions  to  a  common  denominator  — 

1st.   Select  a  convenient  number  for  the  commoii  denominator, 
2d.  Reduce  the  given  fractions  by  the  method  explained  in  A. 

4.  It  will  usually  be  most  convenient  to  select  the  least  com- 
mon multiple  of  the  denominators  of  the  given  fractions  for  a 
common  denominator. 

C.  1.  Reduce  |,  |,  and  \^  to  a  common  deno- 
minator. 

Partial  Solution. — We  first  find  the  least  common  multiple  of  the 
given  denominators  6,  8,  and  12.  It  is  24,  which  we  therefore  select 
for  the  common  denominator.  The  problem  is  now  equivalent  to  the 
following:  *' Reduce  5  |,  and  11  to  twenty-fourths,"  and  may  be 
solved  by  one  of  the  methods  explained  under  A. 

Reduce  the  fractions  in  each  of  the  following 
examples  to  a  common  denominator : — 

2.  i  and  J.  6.  1,  J,  and  |i. 

3.  fandi,  -  7.  /,, -j-V  and  f. 

4.  iand3.  8.  ^^  f,  and /,. 

5.  I  and  |.  9.  -4,  J,  and  ^\. 


I 


LESSON    FIFTY-SEVENTH.  155 


10.  I  J,  and  i. 

11.  |,  i,  and  |. 

12.  |,  I,  and  f 

13.  -rV/p  andi. 
U.  I,  5,  and  i 


15. 

3,  -5j,  and  i. 

16. 

l^iVli^^^di 

17. 

i,  i^iand/,. 

18 

|,  J,  i,  and  /^. 

19. 

iVA^i.^^^il- 

LESSON  LVII. 

A.  In  order  that  fractions  mav  be  added  or  subtracted,  they 
must  be  simple  fractions,  and  have  a  common  denominator. 
Hence  — 

Complex  and  compound  fractions  must  be  reduced  to  simple 
fractions,  and  simple  fractions  to  a  common  denominator,  before 
they  can  be  added  or  subtracted. 

1.  What  is  the  sum  of  |  of  |  +  |f  4-  i  +  j^. 

Solution. — Reducing  the  compound  and  complex  fractions  to  sim- 

pie  ones,  we  have,  |.  of  |  =  f,  and—  =  |.     Hence,  the  problem  be-- 

comes  §  -f-  -|  -f"  t  4"  tV  Selecting  24  as  the  common  denominator, 
and  reducing  the  fractions  to  twenty-fourths,  as  explained  in  Lesson 
LVL,  gives   t  +  1 4-  §  4-  ^7_  _  16  +  |o  ^.  .1 5  +_  14  _2  ij. 

2.  What  is  the  value  of  t^  —  M  of  U  ? 

5^ 

Abbreviated  Solution. — Keducing  the  given  fractions  to  simple  ones 
and  then  to  a  common  denominator,  we  have,— i  —  JLg.  of  11  =  |1  — 

cj  2^  lo  24 

2¥  —  2i 

3.  Add  the  fractions  in  each  of  the  examples  under C,  Lesson 
LVI. 

4.  Find  the  difference  of  the  fractions  in  the  second,  third, 
fourth,  and  fifth  examples  under  C,  Lesson  LVI. 


156  colburm's   first  part. 


5.  In  each  example  following  tlie  fifth  under  C,  Lesson  LVI.. 
subtract  the  last  fraction  from  the  sum  of  the  others. 

6.  l+t?  21.     l+i  +  l? 

7.  2i  +  3i?  22.     f+4  +  i?' 

8.  3J  +  4|?  23.     ^  4-15  +  1? 
9-     7f  +  2§?  24.    ^s,+-;,+|4-i? 

10.  4|-4-7i?  25.  iofj  +  l? 

11.  5i+4§?  26.  |of,?,  +  -t|? 

12.  G  J  —  2t  ?  27.  3i  +  8 J  —  6/^  ? 

13.  7i  — 2J?  ^  28.  4|+ 64—7^3? 

14.  5J+3i?  *  29.  |of|4  +  |ofJ? 

15.  85-8}?  80.    £i+|ofj?,? 

H 

16.  0J-8|?  81.     §+ll+i? 


"7 


17. 


4i  — 2f?  82.  1*   ,    ?f   ,    ?i? 

^  3  +  6i  +   5 

18.  5«  +  2§?  33.  2of«of|i+4? 

19.  7i-4i?  84.  8i  +  34  +  7J? 

20.  9i  +  6J?  35.  ?i  +  |of|? 

^5 

B.  1.  Rufus  bought  a  slate  for  J  of  a  dollar,  a  writing  book 
for  J  of  a  dollar,  a  geography  for  J  of  a  dollar,  and  an  atlas  for 
J,  of  a  dollar.     What  was  the  cost  of  the  whole  ? 

2.  Edward  spends  IJ  hours  each  day  in  studying  history,  1| 
hours  in  studying  geograj^hy,  and  1|-  hours  in  studying  grammar. 
How  many  hours  does  he  spend  in  studying  all  these  branches  ? 

3.  A  man  bought  a  large  pine-apple  and  gave  J  of  it  to  Sarah, 
\  of  it  to  Jane,  ^^  of  it  to  Susan,  JL  of  it  to  Maria,  and  the  rest 
to  Emma.     What  part  of  it  did  he  give  to  Emma? 

4.  I  spent  I  of  my  money  for  land,  j\  of  it  for  buildings,  and 
put  the  rest  at  interest.     What  part  of  it  did  I  put  at  interest? 


LESSON    FIFTY-SEVENTH.  157 


5.  I  bought  an  umbrella  for  $lf,  and  a  pair  of  shoes  for  $2|. 
How  much  did  both  cost  ?  How  much  more  did  the  shoes  cost 
than  the  umbrella  ? 

6.  A  farmer  sold  6J  tons  of  hay,  and  then  had  8|  tons  left. 
How  many  tons  had  he  at  first  ? 

7.  A  man  walked  from  Dedham  to  Boston,  a  distance  of  10 
miles,  in  3  hours.  He  walked  2|  miles  the  first  hour,  and  3y'^j 
miles  the  second.     How  far  did  he  walk  in  the  third  hour  ? 

8.  Mr.  Wheelock  bought  a  book  for  $1|,  and  a  ream  of  paper 
for  $2|,  giving  in  payment  a  five-dollar  bill.  How  much  money 
ought  he  to  receive  back  ? 

9.  Mr.  Nichols's  orchard  contains  3f  acres,  and  his  house-lot 
contains  |  of  an  acre.  How  many  acres  do  both  contain,  and  how 
many  more  acres  are  there  in  his  orchard  than  in  his  house-lot  ? 

10.  Mr.  Turner  bought  4  loads  of  hay.  The  first  weighed  J 
of  a  ton,  the  second  weighed  i  of  a  ton,  the  third  weighed  ^  of  a 
ton,  and  the  fourth  weighed  f  of  a  ton.  What  did  they  all 
weigh  ? 

11.  I  bought  I  of  a  yard  of  silk  velvet  at  $7  per  yard,  and  j 
of  a  yard  of  satin  at  $6  per  yard.     What  did  both  cost  ? 

12.  I  sold  7  barrels  of  apples  at  $2J  per  barrel,  receiving  in 
payment  4  yards  of  cloth  at  $3g  per  yard,  and  the  rest  in  money. 
How  much  money  did  I  receive  ? 

13.  Thil^morning  I  had  $10i,  but  I  have  since  paid  away 
$6/^.     How  much  have  I  left  ? 

14.  A  farmer  gathered  7i  barrels  of  russets,  8|  barrels  of  pip- 
pins, 6-|  barrels  of  greenings,  and  9 J  barrels  of  sweetings.  How 
many  barrels  did  he  gather  in  all  ? 

15.  Arthur,  Richard,  and  Edwin  were  talking  about  their  mo- 
ney. Arthur  said  that  he  had  $4f  <'  Then,"  said  Richard,  "  I 
have  $2-5-  more  than  you  have."    Edwin  thought  a  moment,  and 

14~  '^ 


168  colburn's  first  part. 


then  said,  "  If  I  had  $3 J  more  than  I  now  have,  I  should  have  as 
macn  as  both  of  you  together.  Hov?-  many  did  Richard  have  ? 
How  many  did  Edwin  have  ? 

16.  Rufus  spent  i  of  his  money  for  writing  paper,  J  of  it  for 
pens,  and  the  rest,  which  was  4  cents,  for  a  pencil.  What  part 
of  his  money  did  he  spend  for  a  pencil  ?  How  many  cents  did  he 
spend  in  all,  and  how  many  for  each  article  ? 

17.  A  farmer  has  i  of  his  sheep  in  one  pasture,  |  of  them  in 
another,  and  the  rest,  6  sheep,  in  another.  How  many  has  he  in 
all,  and  how  many  in  each  pasture  ? 

18.  Benjamin  being  asked  his  age,  replied,  <*I  have  spent  j^^ 
of  my  life  in  Brooklyn,  J  of  it  in  New  York,  ^  of  it  in  Baltimore, 
and  the  rest,  6  years,  in  Cincinnati.     What  was  his  age  ? 

19.  A  drover  says  that  if  he  sells  J  of  his  sheep  to  one  man, 
and  J  of  them  to  another,  he  shall  sell  6  more  to  the  first  man 
than  to  the  second.     How  many  sheep  has  he  ? 

20.  James  and  George  were  talking  about  their  ages.  James 
said  that  i  of  his  age  exceeded  J  of  it  by  li  years ;  to  which 
George  replied,  **  Then  you  are  only  |  as  old  as  I  am.'*  What 
was  the  age  of  each  boy  ? 

21.  3i  times  a  certain  number  exceeds  2  times  the  number  by 
12.     What  is  the  number  ? 

22.  If  I  could  sell  my  cow  for  13  dollars  more  than  3  times 
what  she  cost  me,  I  should  receive  $100  for  her.  How^'much  did 
she  cost  me  ? 

23.  There  is  an  orchard  in  which  |  of  the  trees  bear  peaches, 
J  bear  cherries,  J  bear  apples,  and  the  rest  bear  pears.  Now,  if 
there  are  7  more  apple  trees  than  peach  trees,  how  many  trees 
are  there  in  the  orchard,  and  how  many  of  each  kind  ? 

24.  Mr.  Jones  and  Mr.  French  traded  in  company.  Mr.  Jones 
put  in  $3  as  often  as  Mr.  French  put  in  $4.     When  they. came 


LESSON    FIFTY-EIGHTH.  159 

to  divide  the  gain,  it  was  found  that  Mr.  French's  share  was  $8 
more  than  Mr.  Jones's.  How  much  did  they  gain,  and  what  was 
the  share  of  each? 


LESSON  LVIII. 

A.  1.  Of  what  denominations  is  the  number  427 
composed  ? 

Ans. — The  number  427  is  composed  of  4  hundreds,  2  tens,  and 
7  units. 

In  the  same  way  tell  of  what  denominations  each 
of  the  following  numbers  is  composed : — 

2.     678.  5.     5276.  8.     2008. 


3. 

982. 

6. 

3028. 

9. 

3254. 

4. 

201. 

7. 

1406. 

10. 

6897. 

11.  Explain  the  use  of  the  figures  of  the  above 
numbers,  as  in  the  following  — 

Model. — In  427,  the  7  marks  the  units*  place,  and  shows  that 
there  are  7  units ;  tjfie  2  marks  the  tens'  place,  and  shows  that 
there  are  2  tens  ;  the  4  marks  the  hundreds'  place,  and  shows  that 
there  are  4  hundreds. 

12.  Kjive  the  value  of  each  figure  of  the  above 
numbers,  as  in  the  following  — 

Model.  —  In  427,  the  7  =  7  units ;  the  2  =  2  tens,  or  20 
units ;  the  4  =  4  hundreds,  or  40  tens,  or  400  units. 

B      The  foregoing  illustrations  show  that  — 

The  value  of  each  figure  is  ten  times  the  value  it  would  have  if  it 
stood  one  place  farther  to  the  right,  and  one-tenth  of  the  value  it  would 
have  if  it  stood  one  place  farther  to  the  left. 


160  colburn's   first   part. 


==1 


1.   Compare  the  values  expressed  by  the  2's  of  222. 

Ans. — The  first,  or  right  hand  2,  expresses  ^-^  the  value  of  the 
second  2,  and  y|^  the  value  of  the  third  2  ;  the  second,  or  middle 
2,  expresses  10  times  the  value  of  the  first  2,  and  Jg-  the  value 
of  the  third ;  the  third,  or  left-hand  2,  expresses  10  times  the 
value  of  the  second  2,  and  100  times  the  value  of  the  first. 

Compare  in  the  same  way  the  figures  of  each  of 
the  following  numbers: — 

2.     333.  5.     5555.  8.     808. 


3. 

111. 

6. 

9909. 

9. 

7777 

4. 

444. 

7. 

6006. 

10. 

2202 

C.  Marking  the  places  by  a  period,  or  decimal  point  (see 
Lesson  XXII.,  C),  we  may  make  new  places  at  the  right  of  the 
point,  by  calling  the  first  tenths,  the  second  hundredths,  the  third 
thousandths,  &c. 

Thus:  42.37  =  4  tens,  2  units,  3  tenths,  and  7  hundredths; 
.348  =  3  tenths,  4  hundredths,  and  8  thousandths. 

Name  the  denominations  of  the  figures  of  the  fol- 
lowing numbers : — 


1. 

23.47 

4. 

1.46 

7. 

4.596 

10. 

2.7 

2. 

6.825 

5. 

.008 

8. 

1.037 

11. 

4.06 

3. 

.3698 

6. 

.06 

9. 

.027 

12. 

30.03 

D.  Such  numbers  are  read  by  first  reading  the  figures  at  the 
left  of  the  point,  as  though  they  stood  alone, — and  then  reading  the 
figures  at  the  right  of  the  point,  as  though  they  stood  alone^  naming 
afterwards  the  denomination  of  the  right-hand  figure. 

Illustrations. — 42.37  =  42  and  37  one-hundredths  =  42-^-^^. 
.348  =  348  thousandths  =  /_48^,  &c. 


LESSON    FIFTY-EIGHTH.  161 


In  the^same  way,  read  each  of  the  numbers  under  C. 

Numbers  expressed  by  figures  written  both  at  the  right  and 
the  left  of  the  point,  were,  in  the  above  form,  read  as  mixed 
numbers.  They  may,  with  equal  propriety,  be  read  as  improper 
fractions. 

Illustrations.— 42.37  =  42-^^^=  \2_3^7 .  5.5  ,_,  5.^^  _  |.j^  &c. 

Read  each  number  under  C,  which  is  greater  than  1,  as  an 
improper  fraction. 

E.  1.  K  the  decimal  point  of  any  number  be  remored  one  place 
farther  towards  the  right,  or,  which  is  the  same  thing,  the  figures 
be  removed  one  place  towards  the  left,  each  figure  will  represent 
10  times  as  large  a  value  as  before ;  while  if  the  point  be  removed 
one  place  farther  towards  the  left,  or,  which  is  the  same  thing,  the 
figures  be  removed  one  place  toward  the  right,  each  figure  will 
represent  one-tenth  as  large  a  value  as  before. 

2.  A  similar  change  of  two  places,  would  multiply  or  divide  a 
number  by  100, — of  three  places,  by  1000,  &c.,  &;c. 

3.  Hence — To  multiply  a  number  by  10,  it  is  only  necessary  to 
remove  the  point  one  place  farther  towards  the  right;  to  multiply  by 
100,  remove  it  two  places,  ^c,  Sfc. 

4.  So,  to  divide  a  number  by  10,  remove  the  point  one  place  towards 
the  left;  to  divide  by  100,  remove  ii  two  places,  ^c,  ^'c. 

5.  If  there  are  not  figures  enough  at  the  right  or  left  of  the  point 
to  make  these  changes,  annex  or  prefix  zeroes  to  make  up  the  deficiency. 

Illustrations. 

46  X  10  =  460  46  -r  10  =  4.6 

3.7  X  10  =  37  4  -.  10  =  .4 

5.86  X  10  =  58.6         067  -r  10  =  .0067 

234  X  100  =  23400        634  -^  100  =  5.34 
67.8  X  100  =  5780         .8&  -r  100  ==  .0085 
6.294  X  100  =  629.4        6.9  ~  100  =  .069 

14*  L 


162  colburn's  first  part. 


Multiply  each  of  the  following  numbers  by  10  : — 

1.  84  4.     6.24  7.      2847  10.      8246 

2.  5.6  6.     63.7  8.     54.09  11.     .9374 

3.  .63  6.      286  9.     3.275  12.     23.16 

13.  Multiply  each  of  the  above  numbers  by  100,  and  then  by 
1000. 

14.  Divide  each  of  the  above  numbers  by  10,  then  by  100, 
then  by  1000. 

15.  Find  ^>g.  of  each  of  the  above  numbers,  then  ji^,  then 

F.     1.     What  is  .03  of  145.6  ? 

Solution.— Since  .03  of  a  number  equals  3  times  ^J.^  of  that  num- 
ber, it  may  be  found  by  removing  the  point  two  places  further  towards 
the  left  and  multiplying  by  3,  which  would  give  .03  of  145.6  =  3  times 
1.456  =  4.368. 

Note.— Probably  it  will  be  better  to  have  the  pupil  perform  most 


of  these  questions  on 

his  slate. 

2.       .4  of  6.8? 

6.     .07  of  5.6? 

10. 

.003  of  279? 

3.     .05  of  27? 

7.     .02  of  176? 

11. 

.004  of  8.27? 

4.       .3  of  56? 

8.       .2  of  .06? 

12. 

1.2  of  43? 

5.     1.3  of  6.7? 

9.     .25  of  183? 

13. 

1.42  of  .687? 

G.     1.  What  IS  the  quotient  of  4.8  -^  .006? 

Since  the  quotient  of  a  number  divided  by  .006  equals  (Lesson  LV.) 
J  of  1000  times  that  number,  it  may  be  found  by  removing  the  point 
three  places  toward  *he  right  and  dividing  by  6,  which  would  give 
4.8  -f-  .006  =  h  of  4800  =  800. 

2.      4.25  —  5?  6.        32 -r  .008?  10.     .325  —  25? 

3*         25  ~  .05  ?         7.     2.76  ~  1.2  ?  H.       36  -^  .006  ? 

4.  .06  -^  .006  ?       8.^  2.76  -^  .12?  12.       49  -^  4.9  ? 

5,  .0144 -f- .12?         9.     2.76-^.  .012?  13.        37 -r  .037  ? 


LESSON    FIFTY-EIGHTH.  163 


H.  The  term  per  cent  is  often  used  in  place  of  one  hun- 
dredths. 

Thus,  6  per  cent  =  .06,  or  y J^ ;  9  per  cent  =  .09,  or  yf,, 
&c.,  &c. 

1.  I  gathered  43  bushels  of  apples,  receiving  12  per  cent  of 
them  for  my  labor.     How  many  bushels  did  I  receive  ? 

2.  I  bought  a  sleigh  for  $16.20,  and  paid  a  sum  equal  to  8  per 
cent  of  the  cost  for  having  it  repaired.  How  much  did  1  pay  for 
having  it  repaired  ? 

3.  A  man  who  had  87  bushels  of  apples,  sold  .7  of  them  and 
kept  the  rest.  How  many  bushels  did  he  sell  ?  How  many  did 
he  keep  ? 

4.  A  father  left,  at  his  death,  97  acres  of  land,  to  be  so  divided 
that  his  widow  should  have  .4  of  it,  his  oldest  son  .3,  his  youngest 
son  .2,  and  his  daughter  the  rest.     "What  was  the  share  of  each  ? 

5.  George  received  9  per  cent  of  $144,  and  William  received  6 
per  cent  of  $216.     Which  received  the  most? 

6.  A  trader  bought  a  lot  of  goods  for  $36,  and  sold  them  so  as 
to  gain  10  per  cent  of  the  cost.  What  was  his  gain,  and  for  how 
much  did  he  sell  them  ? 

7.  Bought  goods  for  $300,  and  sold  them  so  as  to  gain  15  per 
cent.     What  was  my  gain  ? 

8.  I  gave  $28.60  for  a  lot  of  goods,  but  I  was  obliged  to  sell 
them  so  as  to  lose  8  per  cent.  How  many  dollars  did  I  lose,  and 
for  how  many  dollars  did  I  sell  them  ? 

9.  IMr.  Brown  bought  a  horse  for  $150,  and  sold  him  at  an 
advance,  or  gain,  of  12  per  cent.     What  was  his  gain? 

10.  I  bought  a  carriage  for  $175,  and,  after  paying  12  per 
cent  of  the  cost  for  repairing  it,  I  sold  it  for  $225.  Did  I  gain 
or  lose,  and  how  much  ? 


164  COLBURN*S    FIRST    PART. 


J.  The  money  which  men  charge  for  their  services  in  buying 
or  selling  goods  for  others,  is  called  commission,  and  is  usually  a 
certain  per  cent  of  the  cost  of  the  goods  bought,  and  of  the 
money  received  for  those  sold. 

1.  Mr.  Clarke  sold  a  lot  of  goods  for  Mr.  Davis  for  $500,  at  a 
commission  of  3  per  cent.  What  did  his  commission  amount  to, 
and  how  much  money  would  be  left  for  Mr.  Davis  ? 

2.  I  sold  a  lot  of  goods  for  $250,  at  a  commission  of  4  per  cent. 
What  did  my  commission  amount  to,  and  what  would  be  left  for 
the  owner  of  the  goods  ? 

3.  A  commission  merchant  sold  85  barrels  of  flour,  at  $8  per 
barrel,  receiving  a  commission  of  2  per  cent.  What  was  his  com- 
mission ? 

4.  I  bought  $860  worth  of  cloth  for  Mr.  Arnold,  charging  him 
a  commission  of  2  per  cent.  What  was  my  commission,  and  what 
ought  Mr.  Arnold  to  pay  me  for  the  cloth  and  my  commission  ? 

6.  George  bought  a  jack-knife  for  James  for  75  cents,  charging 
a  commission  of  8  per  cent.  How  much  ought  James  to  pay  for 
the  knife  and  George's  commission  ? 

6.  Mr.  Greene  bought  a  lot  of  shoes  for  Mr.  Gardner,  for  which 
he  paid  $120,  and  charged  3J  per  cent  commission.  What 
ought  Mr.  Gardner  to  pay  for  the  shoes  and  Mr.  Greene's  com- 
mission? 

7.  By  selling  a  horse  for  20  per  cent  more  than  he  cost,  I 
gained  $80.    What  did  he  cost,  and  for  how  much  did  I  sell  him  ? 

Suggestion. —  The  given  per  cent  can  often  be  reduced  to  lower 
terms.     Thus,  20  per  cent.  =  -.?«   =  -l ;  16§  per  cent  =  151  -=  |,  «5;c. 

8.  By  selling  a  lot  of  merchandise  at  an  advance  of  12 J  per 
cent,  I  gained  $9.50.  What  did  it  cost  me,  and  for  how  much 
did  I  sell  it  ? 

9.  My  commission  of  3  per  cent  for  selling  a  lot  of  goods  was 
$15.     For  how  much  did  I  sell  them  ? 


LESSON    i^IFTY-NINTH.  165 

10.  I  lost  25  per  cent  of  the  cost  of  a  horse  by  selling  him  for 
$120.  What  per  cent  of  his  cost  did  I  receive  ?  How  many  dol- 
lars did  he  cost  me  ?     How  many  dollars  did  I  lose  ? 

11.  I  gained  16f  per  cent  of  the  cost  of  a  horse  by  selling  him 
for  $140.     What  was  his  cost,  and  how  many  dollars  did  I  gain? 

12.  By  selling  some  cloth  at  24  cents  per  yard,  I  should  gain  5 
per  cent  more  than  1  should  by  selling  it  at  28  cents  per  yard. 
What  was  its  cost  ? 

13.  By  selling  cloth  at  12 J  cents  per  yard,  I  gain  25  per  cent. 
For  how  much  should  I  sell  it  to  gain  50  per  cent  ? 


LESSON  LIX. 

A.  1.  If  I  should  have  the  use  of  another  man's  horse  for  a 
day,  or  a  week,  I  ought  to  pay  for  it ;  or  if  I  should  occupy  a 
house  or  a  store  belonging  to  another,  I  ought  to  pay  rent  for  the 
use  of  it.  In  like  manner,  if  I  should  borrow  a  sum  of  money,  I 
ought  to  pay  for  the  use  of  it. 

2.  Money  thus  paid  for  the  use  of  money,  is  called  Interest. 

8.  The  money  lent  or  used  is  called  the  Principal,  and  the 
principal  and  interest  together,  form  the  amount. 

Illustrations. — If  I  should  pay  $3  for  the  privilego  of  using  $100 
for  six  months,  the  $3  would  be  the  interest  of  the  $100  for  6  months  ; 
tho  $100  would  be  the  principal,  and  $100  +  $3,  or  $103,  would  be 
the  amount. 

4.  The  interest  is  usually  a  certain  number  of  one  hundredths 
of  the  principal  for  each  year  it  is  used.  This  number  of  one 
hundredths  is  called  the  Rate  per  cent,  or  simply  the  Rate. 

Illustration. — If  a  man  is  to  pay  a  sum  equal  to  -^  -  of  the  prin- 
cipal for  each  year  he  uses  it,  the  rate  is  6  per  cent. 

6.  In  computing  interest,  a  month  is  reckoned  at  30  days. 


166  colburn's  first   part. 


B.  1.  What  is  the  interest  of  $8  for  2  years  9 
mo.,  at  4  per  cent.  ? 

Solution. — At  4  per  cent  per  year,  the  interest  for  2  yr.  9  mo.,  or 
2|  years,  must  be  2|  times  4  per  cent,  or  11  per  cent,  of  the  principal. 
11  per  cent  of  $8  =  11  times  8  cents  =  88  cents,  or  $.88  =  the 
answer. 

What  is  tne  interest  — 

2.  Of  $7  for  2  jr.,  at  6  per  cent  ? 

3.  Of  $9  for  3  yr.,  at  5  per  cent  ? 

4.  Of  $18  for  6  mo.,  at  6  per  cent? 

5.  Of  $248  for  4  mo.,  at  6  per  cent? 

6.  Of  $43.21  for  1  yr.  10  mo.  at  6  per  cent  ? 

7.  Of  $52.30  for  2  yr.  6  mo.,  at  4  per  cent? 

8.  Of  $132  for  1  yr.,  at  7  per  cent? 

9.  Of  $937  for  8  mo.,  at  6  per  cent  ? 

10.  Of  $42.73  for  2  yr.,  at  4 J  per  cent? 

11.  Of  $23.17  for  1  mo.,  at  6  per  cent? 

12.  Of  $24.36  for  9  mo.,  at  8  per  cent? 

13.  Of  $53.27  for  1  yr.  4  mo.,  at  6  per  cent? 

C.  Interest  is  more  frequently  reckoned  at  6  per  cent  per 
year,  than  at  any  other  rate.  Hence,  in  all  the  following  exam- 
ples and  explanations,  interest  should  be  reckoned  at  6  per  cent, 
unless  otherwise  stated. 

2  months  being  J  of  a  year,  interest  for  2  months  at  6  per  cent, 
must  equal  J  of  6  per  cent,  or  1  per  cent  of  the  principal,  which 
may  be  found  by  removing  the  decimal  point  2  places  to  the  left, 
and  is  as  many  cents  as  there  are  dollars  in  the  principal. 

At  6  per  cent  per  year,  what  is  the  interest  for 
2  months  of  — 

1.  $37?  4.     $657?  7.     $85.75? 

2.  $58?  5.     $938?-  8.     $123.79? 

3.  $49?  6.     $8238?  9.     $437.28? 


LESSON    FIFTY-NINTH.  167 


10.  What  is  the  amount  of  each  of  the  above  ? 

J).  Interest  for  2  months  being  1  per  cent  of  the  principal, 
interest  for  100  times  2  months,  or  200  months,  or  1(5  years  8 
months,  must  be  100  per  cent  of  the  principal,  which  is  the  prin- 
cipal itself. 

From  this  we  compute  the  following  table : — 
At  6  per  cent  per  year,  interest  for  — 

200  mo.,  or  16  yr.  8  mo.  =  principal. 

J  of  200  mo.,  or  8  yr.  4  mo.  =  J  of  prin. 

J  of  200  mo.,  or  66f  mo.,  or  5  yr.  (5  mo.  20  da.  =  J  of  prin. 

J  of  200  mo.,  or  50  mo.,  or  4  yr.  2  mo.  =  J  of  prin. 
""  ^  of  200  mo.,  or  40  mo.,  or  3  yr.  4  mo.  =  J  of  prin. 

J  of  200  mo.,  or  33J  mo.,  or  2  yr.  9  mo.  10  da.  =  J  of  prin. 

J  of  200  mo.,  or  25  mo.,  or  2  yr.  1  mo.  =  J  of  prin. 

^i_.  of  200  mo.,  or  20  mo.,  or  1  yr.  8  mo.  =  -yL.  of  prin. 

jL  of  200  mo.,  or  16f  mo.,  or  1  yr.  4  mo.  20  da.  ==  -^L  of  prin. 

j^-g  of  200  mo.,  or  13 J  mo.,  or  1  yr.,  1  mo.  10  da.  =  y^  of  prin, 

j^  of  200  mo.,  or  12i  mo.,  or  1  yr.  15  da.  =  ^'^  of  prin. 


1,  What  is  the  interest  of  $60  for  each  time  men- 
tioned in  the  table  ? 

^n*.  — The  interest  of  $60  for  200  mo.,  or  16  yr.  8  mo.  = 
$60;  for  100  mo.,  or  8  yr.  4  mo.  —  ^  of  $60  =  $30;  for  66| 
mo.,  or  5  yr.  6  mo.  20  da.  =  J  of  $60  =  $20,  &c.,  &c. 

2.  What  is  the  interest  of  $36  for  each  time  men- 
tioned in  the  table  ?     3.  Of  $48.72  ? 


What  is  the  interest  of  — 

4.  $40  for  100  mo.  ?  6.     $64  for  4  yr.  2  mo.  ? 

5.  $48  for  12i  mo.  ?  7.     $24.60  for  5  yr.  6  mo.  20  da.  ? 


168  colburn's  first  part. 


$66  for  16§  mo.  ?  16,  $16.64  for  1  yr.  15  da.  ? 

$24.36  for  66§  mo.  ?  17.  $25.75  for  3  yr.  4  mo.  ? 

$16.98  for  25  mo.  ?  18.  $44.36  for  1  yr.  8  mo.  ? 

$84.60  for  20  mo.  ?  19.  $16.24  for  8  yr.  4  mo.  ? 

$42  for  50  mo.  ?  20.  $44  for  2  yr.  1  mo.  ? 

$37  for  40  mo.  ?  21.  $60.45  for  1  yr.  1  rao.  10  da. : 

$54.72  for  33J  mo.  ?  22.  $43.78  for  16  yr.  8  mo.  ? 

$75.15  for  13 J  mo.  ?  23.  $75  for  2  yr.  9  mo.  10  da.  ? 

24.  What  is  the  amount  of  each  of  the  ahove  ? 


E.  The  interest  for  20  mo.,  or  1  yr.  8rao.,  being  -jJ^  of  the 
principal,  may  be  found  by  removing  the  decimal  point  1  place 
to  the  left,  and  is  as  many  dimes  as  there  are  dollars  in  the 
principal. 

Hence  the  interest  for  — 

J  of  20  mo.,  or  10  mo.,  =  ^  of  ^^  of  principal. 

J  of  20  mo.,  or  6f  mo.,  or  6  mo.  20  da.  =  J  of  ^  of  prin. 

J  of  20  mo.,  or  6  mo.  =  J  of  -J^  of  prin. 

J  of  20  mo.,  or  4  mo.  =  J  of  y'^  of  prin. 

J  of  20  mo.,  or  3 J  mo.,  or  3  mo.  10  da.  =  J  of  ^^  of  prin. 

i  of  20  mo.,  or  2J  mo.,  or  2  mo.  15  da.  =  J  of  ^^  of  prin. 

jJL  of  20  mo.,  or  If  mo.,  or  1  mo.  20  da.  =  ^'^  of  -j'^  of  prin. 

j^  of  20  mo.,  or  IJ  mo.,  or  1  mo.  10  da.  =  -jJ^  of  ^^  of  prin. 

1.  What  is  the  interest  of  $24  for  each  time  men- 
tioned in  the  table  ? 

Ans.— The  interest  of  $24  for  10  mo.  =  J^  of  $2.40  =  $1.20  ; 
for  6|  mo.,  or  6  mo.  20  da.  =  J  of  $2.40  =  $.80,  &c. 

2.  What  is  the   interest  of  $120  for  each  time 


LESSON    FIFTY-NINTH.  169 

mentioned   in   the   table?     3.    Of  $7.50?     4.  Of 
$4.86? 

What  is  the  interest  of — 

5.  $72  for  2 J  mo,  ?  12.  $483.60  for  1  mo.  20  da.  ? 

6.  $60  for  I J  mo.  ?  13.  $27  for  5  mo.  ? 

7.  $486  for  If  mo.  ?  14.  $7.50  for  2  mo.  15  da.  ? 

8.  $15  for  10  mo.  ?  15.  $74.10  for  3  mo.  10  da.  ? 

9.  $2.40  for  6f  mo.  ?  16.  $55  for  1  mo   10  da.  ? 

10.  $64.50  for  4  mo.  ?  17.     $1.86  for  6  mo.  20  da.  ? 

11.  $36.60  for  3J  mo.  ?  18.     $54.20  for  4  mo.  ? 

19.  What  is  the  amount  of  each  of  the  above? 

F.     The  interest  for  2  months,  or  60  days,  "being   ^-J^  of  the 
principal,  it  follows  that  the  interest  for  — 

J  of  2  mo.,  or  1  mo.,  or  30  da.  =  J  of  jj^  of  the  principal. 

J  of  2  mo.,  or  20  da.  =  J  of  ^^^  of  the  prin. 

J  of  2  mo.,  or  15  da.  =  J  of  -j-i.^  of  the  prin. 

J  of  2  mo.,  or  12  da.  =  J  of  y|^  of  the  prin. 

J  of  2  mo.,  or  10  da.  =  J  of  y^^  of  the  prin. 

J^  of  2  mo.,  or  6  da.  =  ^^  of  j^^  or  y^i^^  of  the  prin. 

-jL  of  2  mo.,  or  5  da.  =  ^^  of  j.}^  of  the  prin. 

J  of  6  da.,  or  3  da.  =  J  of  y^*j^  of  the  prin. 

J  of  6  da.,  or  2  da.  =  J  of  y^J^^  of  the  prin. 

I  of  6  da.,  or  1  da.  =  J  of  y^^^^^  of  the  prin. 

1.  What  is  the   interest  of   $432  for  each  time 
mentioned  in  table  ? 

Solution.— The  interest  of  $432  for  2  mo.  is  $4.32;  for  1  mo.  is  I 
of  $4.32,  which  is  $2.16,  &c.,  &c.,  *  *  *  for  6  days,  is  $.432;  for  3 
(lays  is  h  of  $.432,  which  is  $.216,  &c,  &c. 
_ 


170  colburn's  first   part. 


2.  What  is  the  interest  of  $360  for  each  time 
mentioned  in  the  table  ?     3.  Of  $60.30  ? 

What  is  the  interest  of — 

4.  $42  for  20  da.  ?  9.  $192  for  5  da.  ? 

6.  $36.24  for  15  da.  ?  10.  $43.50  for  12  da.  t 

6.  $48  for  10  da.  ?  11.  $86.37  for  30  da. 

7.  $89  for  6  da.  ?  12.  $228  for  1  da.  ? 

8.  $174  for  3  da.  ?  13.  $234  for  2  da.  ? 

14.  What  is  the  amount  of  each  of  the  above  ? 

G.  The  foregoing  principles  furnish  short  and  expeditious 
methods  of  computing  interest  for  any  time  whatever. 

1.  What  is  the  interest  of  $72.60  for  8  mo.  20  da.? 

1st  Solution. — 8  mo.  20  da.  =  6  mo.  20  da.  -f-  2  mo.  The  interest 
of  $72.60  for  6  mo.  20  da.  =  i  of  $7.26  =  $2.42,  and  the  interest  for 
2  mo.  =  $.726,  which,  added  to  $2.42  =  $3,146  =  Ans. 

2d  Solution. — 8  mo.  20  da.  ==  10  mo.  —  1  mo.  10  da.  The  interest 
$72.60  for  10  mo.  =  i  of  $7.26  =  $3.63,  and  the  interest  for  1  mo.  10 
da.  =  ^ij  of  $7.26  =  $.484,  which,  subtracted  from  $3.63  =  $3,146 
=  Ans. 

3d  Solution.  —  8  mo.  20  da.  =  8  mo.  -f-  20  da.  The  interest  of 
$72.60  for  8  mo.  or  4  times  2  mo.  =  4  per  cent  of  $72.60  =  $2,904, 
and  the  interest  for  20  da.  =  J  of  $.726  =  $.242,  which,  added  to 
$2,904  =  $3,146  =  Ans. 

The  work  can  be  wi'itten  as  follows: — 

1st  Solution.  2d  Solution. 

$72.60    =prin.  $72.60    =  prin. 

2A2    =  int.  6  mo.  20  da.  3.63    =  int.  10  mo. 

.726  =  int.  2  mo.  .484  =  int.  1  mo.  10  da. 

$3,146  =  int.  8  mo.  20  da.  $3,146  =  int.  8  mo.  20  da. 

The  form  for  the  third  solution  would  be  similar  to  these. 


LESSON    FIFTY-NINTH.  171 


What  is  the  interest  of  — 

2.     $90  for  3  mo.  16  da.  ?  11.     $54.24  for  7  mo.  15  da.  ? 

3  $128  for  22  mo.  15  da.  ?        12.     $150  for  35  mo.  10  da.  ? 

4  $64  for  2  mo.  10  da.?  13.     $184  for  2  yr.  3  mo.  ? 

5.  $32  for  5  mo.  15  da.  ?  14.  $96  for  52  mo.  15  da.  ? 

6.  $120.90  for  3  mo.  20  da.?  15.  $186.60  for  7  mo.  ? 

7.  $88.24  for  5  mo.  6  da.?  16.  $28.16  for  2  yr.  6  mo.  ? 

8.  $72.96  for  1  mo.  26  da.  ?  17.  $384  for  19  mo.  27  da.? 

9.  $500  for  9  mo.  24  da.  ?  18.  $30.24  for  6  mo.  17  da.? 
10.  $1000  for  3  mo.  29  da.  ?  19.  $450.36  for  3  mo.  15  da.  ? 

H.  Business  men  often  use  such  methods  as  the  following  in 
connexion  with  those  already  explained  : — 

At  6  per  cent,  per  year  the  interest  of  $2  for  1  month  is  1  cent. 
Hence  — 

The  interest  of  $2  is  1  cent  per  month. 

The  interest  of  $20  is  1  dime  per  month. 

The  interest  of  $200  is  1  dollar  per  month. 

1.  What  is  the  interest  of  $2  for  each  of  the  fol- 
lowing times  ? 


3mo.  ? 

2  yr.  3  mo.  ? 

15J  mo.  ? 

9  mo.  ? 

1  yr.  5  mo.  ? 

4  mo.  10  da.  ? 

15  mo.  ? 

2  yr.  IJ  mo.  ? 

2  yr.  7  mo.  ? 

What  is  the  interest  for  each  of  the  above  times 
of  — 

2.  $1?  6.     $5?  10.     $500? 

3.  $6?  7.     $10?  11.     $14? 

4.  $8?  8.     $200?  12.     $80? 

5.  $20?  9.     $50?  13.     $800? 


172  colburn's  first  part. 


At  6  per  cent,  per  year  — 
The  interest  o/  $6  is  1  mill  per  day. 
The  interest  of  $60  is  1  cent  per  day. 
The  interest  o/$600  is  1  dime  per  day. 
The  ifiterest  of  $6000  is  1  dollar  per  day. 

What  is  the  interest  of  $6  for  each  of  the  fol- 
lowing times  ? 

3  da.  ?  1  mo.  3  da.,  or  S3  da.  ?  3  mo.  6  da.  ? 

7  da.  ?  1  mo.  17  da.  ?  6  mo.  12  da.  ? 

19  da.  ?  2  mo.  25  da.  ?  4  mo.  9  da.? 

What  is  the  interest  for  each  of  the  above  times 


of  — 

1.     $60? 

5. 

$600? 

9. 

$6000? 

2.     $30  ? 

6. 

$300? 

10. 

$1000? 

3.     $20? 

7. 

$150? 

11. 

$1500? 

4.     $120? 

8. 

$1800? 

12. 

$500? 

LESSON  LX. 

MISCELLANEOUS   PROBLEMS. 

1.  7  times  8,  plus  4,  divided  by  5,  multiplied  by  3,  minus  4, 
minus  8,  divided  by  2,  divided  by  3,  multiplied  by  9,  multiplied 
by  2  =  how  many  times  ? 

2.  Multiply  J  of  18  by  J  of  8,  add  J  of  27,  divide  by  J  of  28, 
add  ^j  of  33,  and  square  the  number. 

3.  If  I  of  a  yard  of  cloth  worth  14  cents  per  yard,  are  given 
for  §  of  a  pound  of  chocolate,  how  many  pounds  of  coffee  at  12 
cents  per  pound  should  be  given  for  3J  pounds  of  chocolate  ? 


LESSON     SIXTIETH.  173 


4.  If  I  should  expend  the  sum  of  $9  +  $8  +  $5  +  $9  +  $4 
-|-  $7  for  flour  at  $7  per  barrel,  and  sell  the  flour  at  $8  per 
barrel,  then  expend  the  proceeds  for  cloth  at  $3  per  yard,  and 
sell  the  cloth  for  $4  per  yard,  and  then,  after  spending  $10,  and 
losing  $6,  should  expend  the  remainder  for  tea  at  the  rate  of  3 
pounds  for  $2,  how  many  pounds  of  tea  should  I  buy  ? 

5.  Find  the  cost  of  1|  lb.  coffee  at  16  cents  per  lb.,  2i  lb. 
raisins  at  8  cents  per  lb.,  2|  lb.  figs  at  15  cents  per  lb.,  4 J  lb. 
sugar  at  9  cents  per  lb.,  1|  lb.  tea  at  40  cents  per  lb.,  and  J  lb. 
cotton  at  32  cents  per  lb.  Make  out  a  bill  on  the  supposition 
that  you  sold  the  above  articles  to  one  of  your  school-mates. 

6.  What  must  be  the  length  of  the  side  of  a  square  field  con- 
taining J^  as  many  square  rods  as  a  field  9  rods  long  and  8  rods 
wide? 

7.  Arthur  sold  a  certain  number  of  apples  at  the  rate  of  2  for 
a  cent,  and  Robert  sold  as  many  at  the  rate  of  3  for  a  cent. 
Arthur  received  12  cents  more  than  Robert.  How  many  apples 
did  each  sell  ? 

8.  A  thief  drew  J  of  the  wine  out  of  a  certain  cask,  and,  to 
escape  detection,  filled  it  with  water.  The  next  night  he  drew 
out  J  of  the  contents  of  the  cask,  and  again  filled  it  with  water. 
How  many  gills  of  wine  will  there  now  be  in  each  gallon  of  the 
mixture  ? 

9.  2i  times  a  certain  number  added  to  ^  of  that  number  is  5^ 
less  than  3  times  the  number.     What  is  the  number  ? 

10.  A  lady  being  asked  her  age,  replied,  <' My  father  is  30 
years  older  than  my  sister  Sarah,  and  8J  times  the  difference 
between  their  ages  is  5  times  my  father's  age.  Now,  if  you  will 
tell  how  old  my  father  and  sister  are,  I  will  tell  you  how  to  find 
my  age  ?  "  A  correct  answer  having  been  given,  the  lady  said, 
"  To  3  times  my  father's  age,  add  6  times  my  sister's  age,  and 


174  colburn's    first   part. 


you  will  obtain  a  sum  J  of  which  is  9  yeai  s  more  than  4^  times 
my  age  ?"     What  was  the  age  of  each  ? 

11.  A  teacher  wishing  to  obtain  a  bU.<;V  board  15  ft.  long  and 
6  ft.  wide,  bought  boards  for  the  purpose  at  2i  cents  per  square 
foot.  He  hired  a  carpenter  to  make  it,  paying  him  75  cents  for 
his  work.  He  paid  11  cents  per  square  yard  to  have  it  painted 
and  varnished,  and  it  cost  25  cents  to  have  it  brought  to  the 
school-room  and  put  up.     What  was  the  whole  cost  ? 

12.  My  parlor  and  sitting-room  are  each  5  yards  wide,  but 
my  parlor  is  2  yards  longer  than  my  sitting-room.  Yhe  floor  of 
my  sitting-room  contains  30  square  yards.  What  is  the  length 
of  my  parlor  floor,  and  how  many  square  yards  does  it  contain  ? 

13.  David  said  to  Harry,  **  If  i  the  sum  of  our  ages  be  added 
to  i  of  your  age,  the  same  will  equal  |  of  my  age,  and  I  am  12 
years  older  than  you  are.  What  was  the  age  of  each  of  the 
boys? 

14.  Mr.  Warren  bought  a  cask  of  oil  at  $1.20  per  gallon,  but  I 
of  it  leaked  out.  For  how  much  per  gallon  must  he  sell  the  rest 
so  as  neither  to  gain  nor  lose  ? 

15.  Mr.  Allen  owes  Mr.  Mason  62  cents,  and  the  only  coins 
he  has  are  1  half-dollar,  1  quarter-dollar,  1  half-dime,  and  2 
three-cent  pieces,  while  the  only  coins  Mr.  Mason  has  are  4  half- 
dollars,  5  dimes,  and  2  cents.  How  can  change  be  made  so  that 
the  debt  may  be  paid  with  these  coins  ? 

16.  What  number  added  to  J  of  itself  equals  36  more  than  i 
of  the  number  ? 

17.  A  man  sold  6  barrels  of  apples  and  2  barrels  of  pears  for 
$23,  receiving  twice  as  much  per  barrel  for  th,e  pears  as  for  the 
apples.     IIow  many  dollars  did  he  receive  for  each  ? 

18.  By  selling  cloth  at  $3.50  per  yard,  I  lose  12i  per  cent  of 
its  cost.  How  many  dollars  should  I  lose  on  each  yard  by  selling 
it  at  S3  per  yard? 


LESSON     SIXTIETH.  175 


19.  I  sold  i  of  a  lot  of  grain  for  what  |  of  it  cost,  thereby 
gaining  $16..     How  much  did  the  entire  lot  cost  me  ? 

20.  A.  and  B.  traded  in  company.  A.  put  in  $360,  and  B. 
put  in  §  of  J  of  I  of  42  times  i  as  much  as  A.  They  gained  a 
a  sum  equal  to  f  of  their  joint  stock.  How  much  did  they  gain, 
and  what  was  the  share  of  each  ? 

21.  If  Mr.  Walton's  blackboard  were  2  ft.  wider  than  it  now 
is,  it  would  contain  26  more  square  feet,  but  if  it  were  2  feet 
longer,  it  would  contain  11  more  square  feet.  How  many  square 
feet  does  it  contain  ? 

22.  George  has  money  enough  to  buy  2^  quarts  of  chestnuts, 
Rufus  has  twice  as  much  as  George,  and  Edward  has  J  as  much 
as  Rufus.  They  all  have  57  cents.  How  much  are  the  chestnuts 
worth  per  quart,  and  how  many  cents  has  each  of  the  boys  ? 

23.  The  interest  of  Mr.  Butler's  money  for  5  yr.  6  mo.  20  da., 
at  6  per  cent,  will  equal  $8000.     How  much  money  has  he  ? 

24.  If  a  pound  of  rice  is  worth  f  as  much  as  a  pound  of  sugar, 
and  6  lb.  of  rice  and  10  lb.  of  sugar  are  worth  $1.26,  how  much 
are  5  lb.  of  rice  and  7  lb.  of  sugar  worth  ? 

26.  Why  is  it  that  if  we  multiply  any  number  whatever  by  3, 
add  7  to  the  product,  add  the  first  number  taken  to  this,  add  9  to 
this,  divide  this  by  4,  add  3  to  this,  and  then  subtract  from  this 
the  first  number  taken,  the  result  will  always  be  7  ? 

26.  By  selling  cloth  at  $1.25  per  yard,  I  lose  16|  per  cent. 
For  how  much  per  yard  must  I  sell  it  to  gain  20  per  cent  ? 

27.  There  are  f  as  many  acres  in  my  orchard  as  there  are  in 
my  pasture,  and  J  as  many  in  my  garden  as  in  my  orchard.  If 
there  are  17  acres  in  all,  how  many  are  there  in  each  lot? 

28.  I  bought  a  lot  of  goods  for  $600,  and  after  keeping  them 
1  month  17  days,  I  sold  them  for  $650.  Now,  allowing  that  I 
had  to  pay  interest  on  the  money  invested,  at  the  rate  of  6  per 
cent,  what  was  my  net  gain  ? 


176  colburn's   first   part. 


29.  A  man  bought  a  cask  of  wine,  but  §  of  it  leaked  out.  He 
put  in  as  much  water  as  there  was  wine  remaining,  and  sold  the 
mixture  at  the  same  price  per  gallon  that  he  gave  for  it.  What 
part  of  the  cost  did  he  lose  ? 

30.  After  paying  $3  more  than  ^  of  my  money  to  one  man,  and 
§G  more  than  i  of  what  I  had  left  to  another,  I  had  $7  left.  How 
much  did  I  have  at  first  ? 

31.  I  sold  10  bushels  of  corn  for  Mr.  Austin,  and  8  bushels 
for  Mr.  Brown,  receiving  $11  for  the  lot.  Now,  allowing  that 
Mr.  Austin's  corn  is  worth  20  per  cent  more  per  bushel  than  Mr. 
Brown's,  and  that  I  am  to  receive  $1  for  my  services,  how  much 
money  ought  I  to  pay  to  each  ? 


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UNIVERSITY  OF  CALIFORNIA  LIBRARY 


H.  COWPERTHWAIT  &  CO., 

BOOKSELLERS    AND    PUBLISHERS, 

Invite  the  attention  of  the  Pu  lie  to  tb  :  following  f 

VALUABLE  SCHOOL  BOOKS. 

WARREN'S  SERIES  OF  GEOGRAPHIES. 

THE  PRIMAllY  GEOGRAPilY. 

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THE  PMYSICAL  GEOGRAPHY. 

These  three  bo'ks  form  a  complete  geographical  course, 
adapted  to  all  gmdes  of  scliools.  The  series  is  used  in  most 
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GREENE'S     INTRODUCTION     TO    THE    STUDY    OP 

KNuLlSll  (HlAxMMAR. 
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GREENE'S  •  ANALYSIS    OF     THE     ENGLISH    LAN- 
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This  val-jal/ie  scries  of  school  books  was  prepared  by  Prof. 
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they  are  in  general  use  as  text-books  in  the  higher  order  of 
schools  in  all  parts  Of  the  United  States. 


JBER>ilFS  HISTORY  OF  THE  UNITED  STATES. 

This  school  history  is  written  in  a  most  attractive  style; 
and  the  prominent  events  of  our  country's  history  are  pre* 
sented  in  so  pleasing  a  manner  that  the  book  cannot  fail 
greatly  to  interest  and  instruct  the  pupil. 


